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#### Question 1:

Factorize each of the following expressions:

p3 + 27

The given expression to be factorized is

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 2:

y3 + 125

The given expression to be factorized is

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 3:

1 − 27a3

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 4:

8x3y3 + 27a3

The given expression to be factorized is

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 5:

64a3b3

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 6:

$\frac{{x}^{3}}{216}-8{y}^{3}$

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 7:

10x4y − 10xy4

The given expression to be factorized is

Take common from the two terms,. Then we have

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 8:

54x6y + 2x3y4

The given expression to be factorized is

Take common from the two terms,. Then we have

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 9:

32a3 + 108b3

The given expression to be factorized is

Take common from the two terms,. Then we have

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 10:

(a − 2b)3 − 512b3

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 11:

8x2y3 − x5

The given expression to be factorized is

Take common. Then we have

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 12:

1029 − 3x3

The given expression to be factorized is

Take common 3. Then we have from the above expression,

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 13:

x3y3 + 1

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 14:

x4y4xy

The given expression to be factorized is

Take common. Then we have from the above expression,

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 15:

a3 + b3 + a + b

The given expression to be factorized is

This can be written as

=

Recall the formula for sum of two cubes

Using the above formula, we have

Take common. Then we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 16:

Simplify:

(i) $\frac{173×173×173+127×127×127}{173×173-173×127+127×127}$

(ii) $\frac{155×155×155-55×55×55}{155×155+155×55+55×55}\phantom{\rule{0ex}{0ex}}$

(iii) $\frac{1.2×1.2×1.2-0.2×0.2×0.2}{1.2×1.2+1.2×0.2+0.2×0.2}$

(i) The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for sum of two cubes

Using the above formula, the expression becomes

Note that both and b are positive. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

(ii) The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both, b is positive and unequal. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

(iii) The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both, b is positive and unequal. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

#### Question 17:

(a + b)3 − 8(ab)3

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 18:

(x + 2)3 + (x − 2)3

The given expression to be factorized is

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 19:

x6 + y6

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 20:

a12+ b12

The given expression to be factorized is

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 21:

x3 + 6x2 + 12x + 16

The given expression to be factorized is

This can be written as

Take common x2 from first two terms, 2x from the next two terms andfrom the last two terms. Then we have,

Finally, take common. Then we get,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 22:

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula and taking common from the last two terms, we get

Take common. Then we have,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 23:

a3 + 3a2b + 3ab2 + b3 − 8

The given expression to be factorized is

Recall the well known formula

The given expression can be written as

Recall the formula for difference of two cubes

Using the above formula and taking common –2 from the last two terms, we get

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 24:

8a3 b3 − 4ax + 2bx

The given expression to be factorized is

The given expression can be written as

Recall the formula for difference of two cubes

Using the above formula and taking common from the last two terms, we get

Take common. Then we have,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 1:

Factorize:

64a3 + 125b3 + 240a2b + 300ab2

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 2:

125x3 − 27y3 − 225x2y + 135xy2

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is.

#### Question 3:

$\frac{8}{27}{x}^{3}+1+\frac{4}{3}{x}^{2}+2x$

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is.

#### Question 4:

8x3 + 27y3 + 36x2y + 54xy2

The given expression to be factorized is

This can be written in the form

Take common from the last two terms. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is.

#### Question 5:

a3 − 3a2b + 3ab2 b3 + 8

The given expression to be factorized is

This can be written in the form

Take common from the third and fourth terms. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

This can be written in the following form

Recall the formula for the sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is ofis.

#### Question 6:

x3 + 8y3 + 6x2y + 12xy2

The given expression to be factorized is

This can be written in the form

Take common from the last two terms. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is.

#### Question 7:

8x3 + y3 + 12x2y + 6xy2

The given expression to be factorized is

This can be written in the form

Take common 6xy from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is.

#### Question 8:

8a3 + 27b3 + 36a2b + 54ab2

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is.

#### Question 9:

8a3 − 27b3 − 36a2+ 54ab2

The given expression to be factorized is

This can be written in the form

Take common – 18ab from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is ofis.

#### Question 10:

x3 − 12x(x − 4) − 64

The given expression to be factorized is

This can be written in the form

Take common – 12x from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is.

#### Question 11:

a3x3 − 3a2bx2 + 3ab2xb3

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is.

#### Question 1:

Factorize each of the following expressions:

a3 + 8b3 + 64c3 − 24abc

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 2:

x3 − 8y3 + 27z3 + 18xyz

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is ofis.

#### Question 3:

27x3y3z3 − 9xyz

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is .

#### Question 4:

$\frac{1}{27}{x}^{3}-{y}^{3}+125{z}^{3}+5xyz$

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is ofis .

#### Question 5:

8x3 +27y3 − 216z3 + 108xyz

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is ofis .

#### Question 6:

125 + 8x3 − 27y3 + 90xy

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is ofis .

#### Question 7:

8x3 − 125y3 + 180xy + 216

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is ofis .

#### Question 8:

Multiply:
(i) x2 + y2 + z2xy + xz + yz by x + yz

(ii) x2 + 4y2 + z3 + 2xy + xz − 2yz by x − 2yz

(iii) x2 + 4y2 + 2xy − 3x + 6y + 9 by x − 2y + 3

(iv) 9x2 + 25y2 + 15xy + 12x − 20y + 16 by 3x − 5y + 4

(v) x2 + 4y2 + z2 + 2xy + xz – 2yz by (−z + x – 2y)

(v) x2 + 4y2 + z2 + 2xy + xz – 2yz by (−z + – 2y)

$\left({x}^{2}+4{y}^{2}+{z}^{2}+2xy+xz-2yz\right)\left(-z+x-2y\right)\phantom{\rule{0ex}{0ex}}=\left(-z\right)\left({x}^{2}+4{y}^{2}+{z}^{2}+2xy+xz-2yz\right)+\left(x\right)\left({x}^{2}+4{y}^{2}+{z}^{2}+2xy+xz-2yz\right)+\left(-2y\right)\left({x}^{2}+4{y}^{2}+{z}^{2}+2xy+xz-2yz\right)\phantom{\rule{0ex}{0ex}}=-{x}^{2}z-4{y}^{2}z-{z}^{3}-2xyz-x{z}^{2}+2y{z}^{2}+{x}^{3}+4x{y}^{2}+x{z}^{2}+2{x}^{2}y+{x}^{2}z-2xyz-2{x}^{2}y-8{y}^{3}-2y{z}^{2}-4x{y}^{2}-2xyz+4{y}^{2}z\phantom{\rule{0ex}{0ex}}={x}^{3}-8{y}^{3}-{z}^{3}-{x}^{2}z+2{x}^{2}y+{x}^{2}z-2{x}^{2}y-4{y}^{2}z+4x{y}^{2}-4x{y}^{2}+4{y}^{2}z-x{z}^{2}+2y{z}^{2}+x{z}^{2}-2y{z}^{2}-2xyz-2xyz-2xyz\phantom{\rule{0ex}{0ex}}={x}^{3}-8{y}^{3}-{z}^{3}-6xyz$

Hence, the required value is ${x}^{3}-8{y}^{3}-{z}^{3}-6xyz.$

#### Question 9:

(3x − 2y)3 + (2y − 4z)3 + (4z − 3x)3

The given expression to be factorized is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

when.

Using the above formula, the given expression can be written as

Put, and. Then we have

We cannot further factorize the expression.

So, the required factorization is ofis.

#### Question 10:

(2x − 3y)3 + (4z − 2x)3 + (3y − 4z)3

The given expression to be factorized is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the given expression can be written as

Put, and. Then we have

We cannot further factorize the expression.

So, the required factorization is ofis .

#### Question 11:

The given expression to be factorized is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the given expression can be written as

Put, and.

Then we have

We cannot further factorize the expression.

So, the required factorization is of is

#### Question 12:

(a − 3b)3 + (3bc)3 + (ca)3

The given expression to be factorized is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the given expression can be written as

Put, and. Then we have

We cannot further factorize the expression.

So, the required factorization is ofis.

#### Question 13:

$2{\sqrt{2a}}^{3}+3{\sqrt{3b}}^{3}+{c}^{3}-3\sqrt{6}abc$

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is .

#### Question 14:

$3\sqrt{3}{a}^{3}-{b}^{3}-5\sqrt{5}{c}^{3}-3\sqrt{15}abc$

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is.

#### Question 15:

$2\sqrt{2}{a}^{3}+16\sqrt{2}{b}^{3}+{c}^{3}-12abc$

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of is .

#### Question 16:

(x – 2y)3 + (2y – 3z)3 + (3z x)3

#### Question 17:

Find the value of x3 + y3 − 12xy + 64, when x + y =−4

The given expression is

It is given that

The given expression can be written in the form

Recall the formula

Using the above formula, we have

#### Question 18:

If a, b, c are all non-zero and a + b + c = 0, prove that $\frac{{a}^{2}}{bc}+\frac{{b}^{2}}{ca}+\frac{{c}^{2}}{ab}=3$.

#### Question 1:

The factors of x3x2yxy2 + y3 are

(a)(x + y) (x2xy + y2)

(b) (x + y) (x2 + xy + y2)

(c) (x + y)2 (xy)

(d) (x − y)2 (x + y)

The given expression to be factorized is

Take common from the first two terms and from the last two terms. That is

Finally, take commonfrom the two terms. That is

So, the correct choice is (d).

#### Question 2:

The factors of x3 − 1 + y3 + 3xy are

(a) (x − 1 + y) (x2 + 1 + y2 + x + yxy)

(b) (x + y + 1) (x2 + y2 + 1 −xyx y)

(c) (x − 1 + y) (x2 − 1 − y2 + x + y + xy)

(d) 3(x + y −1) (x2 + y2 − 1)

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

So, the correct choice is (a).

#### Question 3:

The factors of 8a3 + b3 − 6ab + 1 are

(a) (2a + b − 1) (4a2 + b2 + 1 − 3ab − 2a)

(b) (2ab + 1) (4a2 + b2 − 4ab + 1 − 2a + b)

(c) (2a + b + 1) (4a2 + b2 + 1 −2abb − 2a)

(d) (2a − 1 + b) (4a2 + 1 − 4ab − 2ab)

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

So, the correct choice is (c).

#### Question 4:

(x + y)3 − (x − y)3 can be factorized as

(a) 2y (3x2 + y2)

(b) 2x (3x2 + y2)

(c) 2y (3y2 + x2)

(d) 2x (x2+ 3y2

The given expression to be factorized is

Recall the formula for difference of two cubes

Using the above formula, we have,

So, the correct choice is (a).

#### Question 5:

The expression (ab)3 + (b c)3 + (ca)3 can be factorized as

(a) (ab) (b c) (ca)

(b) 3(ab) (bc) (ca)

(c) −3(ab) (bc) (ca)

(d) (a + b + c) (a2 + b2 + c2abbcca)

The given expression is

Let, and. Then the given expression becomes

Note that:

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

So, the correct choice is (b).

#### Question 6:

The value of $\frac{\left(2.3{\right)}^{3}-0.027}{\left(2.3{\right)}^{2}+0.69+0.09}$

(a) 2

(b) 3

(c) 2.327

(d) 2.273

The given expression is

This can be written in the form

Assumeand. Then the given expression can be rewritten as

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both a and b are positive, unequal. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

So, the correct choice is (a).

#### Question 7:

The value of $\frac{\left(0.013{\right)}^{3}+\left(0.007{\right)}^{3}}{\left(0.013{\right)}^{2}-0.013×0.007+\left(0.007{\right)}^{2}}$ is

(a) 0.006

(b) 0.02

(c) 0.0091

(d) 0.00185

The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for sum of two cubes

Using the above formula, the expression becomes

Note that both and b are positive. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

So, the correct choice is (b).

#### Question 8:

Mark the correct alternative in each of the following:

The factors of a2 − 1 − 2xx2 are

(a) (a − x + 1) (a − x − 1)

(b) (a + x − 1) (a − x + 1)

(c) (a + x +1) (a − x + 1)

(d) none of these

The given expression to be factorized is

Take commonfrom the last three terms and then we have

So, the correct choice is (c).

#### Question 9:

The factors of x4 + x2 + 25 are

(a) (x2 + 3x + 5) (x2 − 3x + 5)

(b) (x2 + 3x + 5) (x2 + 3x − 5)

(c) (x2 + x +5) (x2x + 5)

(d) none of these

The given expression to be factorized is

This can be written in the form

So, the correct choice is (a).

#### Question 10:

The factors of x2 + 4y2 + 4y − 4xy − 2x − 8 are

(a) (x − 2y −4) (x − 2y + 2)

(b) (xy + 2) (x − 4y − 4)

(c) (x + 2y − 4) (x + 2y + 2)

(d) none of these

The given expression to be factorized is

This can be arrange in the form

Let. Then the above expression becomes

Put.

So, the correct choice is (a).

#### Question 11:

The factors of x3 − 7x + 6 are

(a) x (x − 6) (x − 1)

(b) (x2 − 6) (x − 1)

(c) (x + 1) (x + 2) (x + 3)

(d) (x − 1) (x + 3) (x − 2)

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms. Then we have

Finally, take commonfrom the above expression,

So, the correct choice is (d).

#### Question 12:

The expression x4 + 4 can be factorized as

(a) (x2 + 2x + 2) (x2 − 2x + 2)

(b) (x2 + 2x + 2) (x2 + 2x + 2)

(c) (x2 − 2x − 2) (x2 − 2x + 2)

(d) (x2 + 2) (x2 − 2)

The given expression to be factorized is

This can be written in the form

So, the correct choice is (a).

#### Question 13:

If 3x = a + b + c, then the value of (xa)3 + (xb)3 + (xc)3 − 3(xa) (xb) (xc) is

(a) a + b + c

(b) (ab) (bc) (ca)

(c) 0

(d) none of these

The given expression is

Recall the formula

Using the above formula the given expression becomes

Given that

Therefore the value of the given expression is

So, the correct choice is (c).

#### Question 14:

If (x + y)3 − (xy)3 − 6y(x2y2) = ky2, then k =

(a) 1

(b) 2

(c) 4

(d) 8

The given equation is

Recall the formula

Using the above formula, we have

, provided.

So, the correct choice is (d).

#### Question 15:

If x3 − 3x2 + 3x − 7 = (x + 1) (ax2 + bx + c), then a + b + c =

(a) 4

(b) 12

(c) −10

(d) 3

The given equation is

x3 − 3x2 + 3x − 7 = (x + 1) (ax2 + bx + c)

This can be written as

${x}^{3}-3{x}^{2}+3x-7=\left(x+1\right)\left(a{x}^{2}+bx+c\right)\phantom{\rule{0ex}{0ex}}⇒{x}^{3}-3{x}^{2}+3x-7=a{x}^{3}+b{x}^{2}+cx+a{x}^{2}+bx+c\phantom{\rule{0ex}{0ex}}⇒{x}^{3}-3{x}^{2}+3x-7=a{x}^{3}+\left(a+b\right){x}^{2}+\left(b+c\right)x+c$

Comparing the coefficients on both sides of the equation.

We get,

c = -7 .......(4)

Putting the value of a from (1) in (2)

We get,

So the value of a, b and c is 1, – 4 and -7 respectively.

Therefore,

a + b + c =1 - 4 - 7 = -10

So, the correct choice is (c).

#### Question 1:

The factorized form of the expression y2 + (x – 1)y x is ____________.

${y}^{2}+\left(x-1\right)y-x\phantom{\rule{0ex}{0ex}}={y}^{2}+xy-y-x\phantom{\rule{0ex}{0ex}}=y\left(y+x\right)-1\left(y+x\right)\phantom{\rule{0ex}{0ex}}=\left(y-1\right)\left(y+x\right)$

Hence, the factorized form of the expression y2 + (x – 1)– x is (y â€‹– 1)(y + x).

#### Question 2:

The factorized form of a3 + (ba)3b3 is ____________.

Hence, the factorized form of a3 + (b – a)3 – b3 is 3ab(a – b).

If

Hence, if

#### Question 4:

Factorization of the polynomial $11{x}^{2}-10\sqrt{3}x-3$ gives ____________.

$11{x}^{2}-10\sqrt{3}x-3\phantom{\rule{0ex}{0ex}}=11{x}^{2}-11\sqrt{3}x+\sqrt{3}x-3\phantom{\rule{0ex}{0ex}}=11x\left(x-\sqrt{3}\right)+\sqrt{3}\left(x-\sqrt{3}\right)\phantom{\rule{0ex}{0ex}}=\left(11x+\sqrt{3}\right)\left(x-\sqrt{3}\right)$

Hence, factorization of the polynomial $11{x}^{2}-10\sqrt{3}x-3$ gives $\overline{)\left(11x+\sqrt{3}\right)\left(x-\sqrt{3}\right)}$.

#### Question 5:

The polynomial x2 + y2z2 – 2xy on factorization gives _____________.

Hence, the polynomial x2 + y2 – z2 – 2xy on factorization gives $\overline{)\left(x-y+z\right)\left(x-y-z\right)}$.

#### Question 6:

The factors of the expression $a+b+c+2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca}$ are ____________.

Hence, the factors of the expression $a+b+c+2\sqrt{ab}-2\sqrt{bc}-2\sqrt{ca}$ are .

#### Question 7:

The polynomial , x6 + 64y6 on factorization gives _____________.

Hence, the polynomial x6 + 64y6 on factorization gives $\overline{)\left({x}^{2}+4{y}^{2}\right)\left({x}^{4}+16{y}^{4}-4{x}^{2}{y}^{2}\right)}$.

#### Question 8:

The factorization form of a4 + b4 a2b2 is _____________.

Hence, the factorization form of a4 + b– a2b2 is $\overline{)\left({a}^{2}+{b}^{2}+\sqrt{3}ab\right)\left({a}^{2}+{b}^{2}-\sqrt{3}ab\right)}$.

#### Question 9:

If , then the value of $27{x}^{3}-\frac{{y}^{3}}{125}$ is ____________.

Hence, the value of $27{x}^{3}-\frac{{y}^{3}}{125}$ is 1090.

#### Question 10:

The factorized form of $\frac{1}{xyz}\left({x}^{2}+{y}^{2}+{z}^{2}\right)+2\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$ is _____________.

Hence, the factorized form of $\frac{1}{xyz}\left({x}^{2}+{y}^{2}+{z}^{2}\right)+2\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$ is $\overline{)\frac{1}{xyz}{\left(x+y+z\right)}^{2}}.$

#### Question 11:

The factorized form of a3 + b3 + 3ab – 1 is ____________.

Hence, the factorized form of a3 + b3 + 3ab – 1 is  $\overline{)\left(a+b-1\right)\left({a}^{2}+{b}^{2}+1-ab+b+a\right)}.$

#### Question 1:

If a + b + c = 0, then write the value of a3 + b3 + c3.

Recall the formula

When, we have

#### Question 2:

If a2 + b2 + c2 = 20 and a + b + c = 0, find ab + bc + ca.

Recall the formula

Given that

Then we have

#### Question 3:

If a + b + c = 9 and ab + bc + ca = 40, find a2 + b2 +c2.

Recall the formula

Given that

,

Then we have

#### Question 4:

If a2 + b2 + c2 = 250 and ab + bc + ca = 3, find a + b + c.

Recall the formula

Given that

,

Then we have

#### Question 5:

Write the value of 253 − 753 + 503.

The given expression is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

#### Question 6:

Write the value of 483 − 303 − 183.

The given expression is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

#### Question 7:

Write the value of ${\left(\frac{1}{2}\right)}^{3}+{\left(\frac{1}{3}\right)}^{3}-{\left(\frac{5}{6}\right)}^{3}.$

The given expression is

Let, and. Then the given expression becomes

Note that:

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

#### Question 8:

Write the value of 303 + 203 − 503.

The given expression is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

#### Question 9:

Factorize: x4 + x2 + 25.

The given expression to be factorized is

This can be written in the form

We cannot further factorize the expression.

So, the required factorization is.

#### Question 10:

Factorize : x2 − 1 − 2aa2

The given expression to be factorized is

Take commonfrom the last three terms and then we have

We cannot further factorize the expression.

So, the required factorization is.

#### Question 1:

Factorize:

1.

The given expression to be factorized is

Take common x from the first two terms and -3 from the last two terms. That is

Finally, take common x2 + 1from the two terms. That is

We cannot further factorize the expression.

So, the required factorization is.

#### Question 2:

Factorize:

2. a(a+b)3 − 3a2b (a + b)

The given expression to be factorized is

Take common from the two terms. That is

Expand the term within the second braces.

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 3:

Factorize:

3.

The given expression to be factorized is

We know that

The given expression then becomes

Take common from the two terms. That is

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 4:

Factorize:

a2x2 + (ax2 + 1)x + a

The given expression to be factorized is

Simplify the middle term. That is

Take common from the first two terms and 1 from the last two terms. That is

Finally, take commonfrom the two terms. That is

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 5:

Factorize:
x2 + y − xy − x

The given expression to be factorized is

Rearrange the given expression as

Take common x from the first two terms and -1 from the last two terms. That is

Finally, take commonfrom the two terms. That is

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 6:

Factorize:

x3 − 2x2y + 3xy2 − 6y3

The given expression to be factorized is

Take common from the first two terms and from the last two terms. That is

Finally, take commonfrom the two terms. That is

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 7:

Factorize:

6ab − b2 + 12ac − 2bc

The given expression to be factorized is

Take common b from the first two terms and  from the last two terms. That is

Finally, take commonfrom the two terms. That is

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 8:

Factorize:

The given expression to be factorized is

Take common 4 from the last two terms. That is

Again take commonfrom the two terms of the above expression.

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 9:

Factorize:

(a − b + c)2 + (b − c + a)2 + 2(a − b + c) (b − c + a)

The given expression to be factorized is

This can be written as

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 10:

Factorize:

a2 + 2ab +b2c2

The given expression to be factorized is

This can be arrange in the form

Substitutingin the above expression, we get.

Put.

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 11:

Factorize:

a2 + 4b2 − 4ab − 4c2

The given expression to be factorized is

This can be arrange in the form

Substitute.

Put.

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 12:

Factorize:

x
2y2 − 4xz + 4z2

The given expression to be factorized is

Rearrange the terms as

Substitutingin the avove expression,

Put.

We cannot further factorize the expression.

So, the required factorization ofis.

#### Question 13:

Factorize:

$2{x}^{2}-\frac{5}{6}x+\frac{1}{12}$

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 14:

Factorize:

${x}^{2}+\frac{12}{35}x+\frac{1}{35}$

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 15:

Factorize:

$21{x}^{2}-2x+\frac{1}{21}$

The given expression to be factorized is

This can be written in the form

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 16:

Give possible expressions for the length and breadth of the rectangle having 35y2 + 13y − 12 as its area.

The area of the rectangle is

First we will factorize the above expression. This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

The area of a rectangle having length a and breadth bis ab.

Here we don’t know the bigger or the smaller factor. So, the two possibilities are

(ii) Length is and breadth is

#### Question 17:

What are the possible expressions for the dimensions of the cuboid whose volume is 3x2− 12x.

The volume of the cuboid is

First we will factorize the above expression.

Take commonfrom the two terms of the above expression,

The volume of a cuboid having length, breadth b and height is.

Here the word ‘dimensions’ stands for the length, breadth and height of the cuboid. So, the three possibilities are

(i) Length is, breadth is x and height is

(ii) Length is x , breadth isand height is

(iii) Length is, breadth isand height is x

There are many other possibilities also, because we can consider the product of two simple factors as a single factor.

#### Question 18:

Factorize:

The given expression to be factorized is

We have

Use the above result in the original expression to get

Substituting in the above , we get

Put.

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 19:

Factorize:

(x+2) (x2+25) − 10x2 − 20x

The given expression to be factorized is

Take common from the last two terms. That is

Again take commonfrom the two terms of the above expression. Then

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 20:

Factorize:

The given expression to be factorized is

This can be written in the form

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 21:

Factorize:

a2 + b2 + 2(ab + bc + ca)

The given expression to be factorized is

This can be written as

Take commonfrom the last two terms.

Finally, take common from the two terms of the above expression.

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 22:

Factorize:

4(x − y)2 − 12(x − y) (x + y) + 9(x + y)2

The given expression to be factorized is

Substituting andin the above expression, we get

=

This can be arrange in the form

Putand.

Take common -1 from the expression within the braces.

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 23:

Factorize:
a2 + b2 + 2bc − c2

The given expression to be factorized is

This can be arrange in the form

Substitutingin the above expression, we get.

Put.

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 24:

Factorize:

xy
9yx9

The given expression to be factorized is

This can be written in the form

Take commonfrom the two terms of the above expression

We cannot further factorize the expression.

So, the required factorization of is

#### Question 25:

Factorize:

x4 + x2y2 + y4

The given expression to be factorized is

Add and subtract the termin the given expression.

Substitutingin the above expression, we get

Putin the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 26:

Factorize:

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms.

Finally, take commonfrom the above expression. Then we have

We cannot further factorize the expression.

So, the required factorization is.

#### Question 27:

Factorize:

${x}^{2}-2\sqrt{2}x-30$

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 28:

Factorize:

${x}^{2}-\sqrt{3}x-6$

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms. Then we have

Finally take commonfrom the above expression. Then we have

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 29:

Factorize:

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms. Then we have

Finally take commonfrom the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 30:

Factorize:

${x}^{2}+2\sqrt{3}x-24$

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 31:

Factorize:

$5\sqrt{5}{x}^{2}+20x+3\sqrt{5}$

The given expression to be factorized is

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 32:

Factorize:

$2{x}^{2}+3\sqrt{5}x+5$

The given expression to be factorized is

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 33:

Factorize:

9(2ab)2 − 4(2ab) − 13

The given expression to be factorized is

Substitutingin the above expression, we get

This can be written in the form

Take common x from the first two terms and 1 from the last two terms,

Finally take commonfrom the above expression,

Put,

We cannot further factorize the expression.

So, the required factorization ofis.

#### Question 34:

Factorize:

7(x − 2y)2 − 25(x − 2y) + 12

The given expression to be factorized is

Substitutingin the above expression, we get

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

Put in the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

#### Question 35:

Factorize:

2(x + y)2 − 9(x + y) − 5

The given expression to be factorized is

Substitutingin the above expression, we get

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

Put. Then we have

We cannot further factorize the expression.

So, the required factorization of is.

View NCERT Solutions for all chapters of Class 9
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