What is the coefficient of $x^3$ when $$x^4-3x^3 + 5x^2-6x + 1$$is multiplied by $$2x^3 - 3x^2 + 4x + 7$$and the like terms are combined?
Instead of expanding the entire product, we can look only at terms that will multiply to give $x^3$.  We know that: $$x^3=x^3\cdot 1=x^2\cdot x=x\cdot x^2=1\cdot x^3$$Knowing this, the $x^3$ term in the expansion will be the sum of these four terms: $$(-3x^3)(7)+(5x^2)(4x)+(-6x)(-3x^2)+(1)(2x^3)$$We simplify to find: \begin{align*}
&(-3x^3)(7)+(5x^2)(4x)+(-6x)(-3x^2)+(1)(2x^3)\\
&\qquad=-21x^3+20x^3+18x^3+2x^3\\
&\qquad=\boxed{19}x^3.
\end{align*}
19
Level 4
Algebra
Problem: What is the coefficient of $x^3$ when $$x^4-3x^3 + 5x^2-6x + 1$$is multiplied by $$2x^3 - 3x^2 + 4x + 7$$and the like terms are combined?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the coefficient of \(x^3\) in the product of the two polynomials, we need to consider how \(x^3\) can be formed from the terms of both polynomials. 

The first polynomial is \(x^4 - 3x^3 + 5x^2 - 6x + 1\) and the second polynomial is \(2x^3 - 3x^2 + 4x + 7\). 

We will identify the combinations of terms from each polynomial that will yield \(x^3\) when multiplied together. 

1. The term \(x^4\) from the first polynomial cannot contribute to \(x^3\) since it would require a term of degree \(-1\) from the second polynomial, which does not exist.

2. The term \(-3x^3\) from the first polynomial can be multiplied by the constant term \(7\) from the second polynomial:
   \[
   -3x^3 \cdot 7 = -21x^3
   \]

3. The term \(5x^2\) from the first polynomial can be multiplied by the \(4x\) term from the second polynomial:
   \[
   5x^2 \cdot 4x = 20x^3
   \]

4. The term \(-6x\) from the first polynomial can be multiplied by the \(-3x^2\) term from the second polynomial:
   \[
   -6x \cdot -3x^2 = 18x^3
   \]

5. The constant term \(1\) from the first polynomial can be multiplied by the \(2x^3\) term from the second polynomial:
   \[
   1 \cdot 2x^3 = 2x^3
   \]

Now, we will sum all the contributions to the coefficient of \(x^3\):
\[
-21 + 20 + 18 + 2
\]

Calculating this step-by-step:
- Start with \(-21 + 20 = -1\)
- Then, \(-1 + 18 = 17\)
- Finally, \(17 + 2 = 19\)

Thus, the coefficient of \(x^3\) in the product of the two polynomials is \(19\).<|END_OF_LATENT|> The final answer is $19$.