To multiply the polynomials \( (2x - 7) \) and \( (3x^2 + 7x - 8) \), we can use the distributive property (also known as the FOIL method for binomials). Let's carry out the multiplication step by step:
\[
(2x - 7)(3x^2 + 7x - 8)
\]
Distributing \( 2x \) to each term in \( (3x^2 + 7x - 8) \):
\[
2x \cdot 3x^2 = 6x^3
\]
\[
2x \cdot 7x = 14x^2
\]
\[
2x \cdot (-8) = -16x
\]
Distributing \( -7 \) to each term in \( (3x^2 + 7x - 8) \):
\[
-7 \cdot 3x^2 = -21x^2
\]
\[
-7 \cdot 7x = -49x
\]
\[
-7 \cdot (-8) = 56
\]
Now combine all the results:
\[
6x^3 + 14x^2 - 16x - 21x^2 - 49x + 56
\]
Next, we consolidate the like terms:
- The \( x^2 \) terms: \( 14x^2 - 21x^2 = -7x^2 \)
- The \( x \) terms: \( -16x - 49x = -65x \)
So we can write the result in descending order:
\[
6x^3 - 7x^2 - 65x + 56
\]
Now that we have the polynomial, we can answer the questions about the coefficients:
1. The value of the coefficient of the \( x^3 \) term is **6**.
2. The value of the coefficient of the \( x^2 \) term is **-7**.
3. The value of the coefficient of the \( x \) term is **-65**.
4. The value of the coefficient of the constant term is **56**.
Here are the responses summarized:
- The value of the coefficient of the \( x^3 \) is **6**.
- The value of the coefficient of the \( x^2 \) is **-7**.
- The value of the coefficient of the \( x \) is **-65**.
- The value of the coefficient of the constant is **56**.