Question

Expand the expression to a polynomial in standard form:
left parenthesis, minus, x, plus, 4, right parenthesis, left parenthesis, 3, x, squared, minus, 2, x, minus, 7, right parenthesis
(−x+4)(3x
2
−2x−7)

Answer 1

To expand the expression \((-x + 4)(3x^2 - 2x - 7)\), we can use the distributive property (also known as the FOIL method for binomials). We will multiply each term in the first binomial by each term in the second polynomial.

1. **Multiply \(-x\) by each term in the second polynomial:**
- \(-x \cdot 3x^2 = -3x^3\)
- \(-x \cdot (-2x) = 2x^2\)
- \(-x \cdot (-7) = 7x\)

So, multiplying \(-x\) gives us:
\[
-3x^3 + 2x^2 + 7x
\]

2. **Multiply \(4\) by each term in the second polynomial:**
- \(4 \cdot 3x^2 = 12x^2\)
- \(4 \cdot (-2x) = -8x\)
- \(4 \cdot (-7) = -28\)

So, multiplying \(4\) gives us:
\[
12x^2 - 8x - 28
\]

3. **Combine all the results:**
Combine the results from the two steps:

\[
(-3x^3 + 2x^2 + 7x) + (12x^2 - 8x - 28)
\]

4. **Combine like terms:**
- The \(x^3\) term: \(-3x^3\)
- The \(x^2\) terms: \(2x^2 + 12x^2 = 14x^2\)
- The \(x\) terms: \(7x - 8x = -x\)
- The constant term: \(-28\)

Putting this all together, we have:

\[
-3x^3 + 14x^2 - x - 28
\]

Thus, the expression \((-x + 4)(3x^2 - 2x - 7)\) in standard polynomial form is:

\[
\boxed{-3x^3 + 14x^2 - x - 28}
\]