To solve the problem of finding [tex]\((k \cdot p)(x)\)[/tex] given the functions [tex]\(k(x) = 2x^2 - 7\)[/tex] and [tex]\(p(x) = x - 4\)[/tex], follow these step-by-step instructions:
1. Write down the given functions:
- [tex]\(k(x) = 2x^2 - 7\)[/tex]
- [tex]\(p(x) = x - 4\)[/tex]
2. Form the product [tex]\((k \cdot p)(x)\)[/tex]:
[tex]\[
(k \cdot p)(x) = k(x) \cdot p(x)
\][/tex]
Substitute the given functions for [tex]\(k(x)\)[/tex] and [tex]\(p(x)\)[/tex]:
[tex]\[
(k \cdot p)(x) = (2x^2 - 7)(x - 4)
\][/tex]
3. Distribute the terms:
To expand [tex]\((2x^2 - 7)(x - 4)\)[/tex], use the distributive property (also known as the FOIL method):
[tex]\[
(2x^2 - 7)(x - 4) = 2x^2 \cdot x + 2x^2 \cdot (-4) - 7 \cdot x - 7 \cdot (-4)
\][/tex]
4. Multiply each term:
- [tex]\(2x^2 \cdot x = 2x^3\)[/tex]
- [tex]\(2x^2 \cdot (-4) = -8x^2\)[/tex]
- [tex]\(-7 \cdot x = -7x\)[/tex]
- [tex]\(-7 \cdot (-4) = 28\)[/tex]
5. Combine all these terms:
Put all the terms together:
[tex]\[
(k \cdot p)(x) = 2x^3 - 8x^2 - 7x + 28
\][/tex]
Therefore, the correct result for [tex]\((k \cdot p)(x)\)[/tex] is [tex]\(2x^3 - 8x^2 - 7x + 28\)[/tex].
Looking at the provided possible answers, none of them are correct, based on the computed expression. Therefore, as none of the provided options matches [tex]\(2x^3 - 8x^2 - 7x + 28\)[/tex], you might want to reconsider the given choices in context. The correct result derived from the given [tex]\(k(x)\)[/tex] and [tex]\(p(x)\)[/tex] is:
[tex]\[
2x^3 - 8x^2 - 7x + 28
\][/tex]
