To find the product of the two given functions [tex]\( F(x) = x - 4 \)[/tex] and [tex]\( G(x) = x - 6 \)[/tex], follow these steps:
1. Write down the expressions for the functions:
- [tex]\( F(x) = x - 4 \)[/tex]
- [tex]\( G(x) = x - 6 \)[/tex]
2. Multiply the expressions together:
We need to find [tex]\( F(x) \cdot G(x) \)[/tex]:
[tex]\( (x - 4)(x - 6) \)[/tex]
3. Distribute each term in the first expression by each term in the second expression. This means you multiply each pair of terms one-by-one:
[tex]\[
\begin{aligned}
(x - 4)(x - 6) &= x \cdot x + x \cdot (-6) + (-4) \cdot x + (-4) \cdot (-6) \\
&= x^2 - 6x - 4x + 24 \\
&= x^2 - 10x + 24.
\end{aligned}
\][/tex]
4. Combine like terms:
Simplify the expression by combining the linear terms ([tex]\(-6x\)[/tex] and [tex]\(-4x\)[/tex]):
[tex]\[
x^2 - 10x + 24.
\][/tex]
So, the product of the functions [tex]\( F(x) = x - 4 \)[/tex] and [tex]\( G(x) = x - 6 \)[/tex] is:
[tex]\[ (x - 4)(x - 6) = x^2 - 10x + 24 \][/tex]
This polynomial [tex]\( x^2 - 10x + 24 \)[/tex] is the final product of the two given functions.
