To solve for [tex]\((k \cdot p)(x)\)[/tex] given the functions [tex]\(k(x) = 2x^2 - 7\)[/tex] and [tex]\(p(x) = x - 4\)[/tex], you need to find the product of these two functions.
First, let's write the expression for [tex]\((k \cdot p)(x)\)[/tex]:
[tex]\[ (k \cdot p)(x) = k(x) \cdot p(x) \][/tex]
Substitute the given functions into the expression:
[tex]\[ (k \cdot p)(x) = (2x^2 - 7) \cdot (x - 4) \][/tex]
Next, expand the product using the distributive property (also known as FOIL method for binomials):
[tex]\[
\begin{align*}
(k \cdot p)(x) &= (2x^2 - 7)(x - 4) \\
&= 2x^2 \cdot x + 2x^2 \cdot (-4) + (-7) \cdot x + (-7) \cdot (-4) \\
&= 2x^3 - 8x^2 - 7x + 28
\end{align*}
\][/tex]
So, the expanded form of [tex]\((k \cdot p)(x)\)[/tex] is:
[tex]\[ (k \cdot p)(x) = 2x^3 - 8x^2 - 7x + 28 \][/tex]
Notice that none of the provided options match this polynomial exactly. This implies there might be a mistake in the problem options. However, considering the expanded form from our calculations, none of the provided answers are correct. Therefore, the accurate expression for [tex]\((k \cdot p)(x)\)[/tex] based on our step-by-step calculation is:
[tex]\[ (k \cdot p)(x) = 2x^3 - 8x^2 - 7x + 28 \][/tex]
