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Answer :
To solve the problem of finding [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the two given functions, [tex]\( f(x) = -5x \)[/tex] and [tex]\( g(x) = 8x^2 - 5x - 9 \)[/tex]. We'll perform the multiplication step by step to reach our final expression.
Here's how you would approach it:
1. Distribute [tex]\( f(x) \)[/tex] over each term in [tex]\( g(x) \)[/tex]:
- Multiply [tex]\(-5x\)[/tex] by the first term in [tex]\( g(x) \)[/tex], which is [tex]\( 8x^2 \)[/tex]:
[tex]\[
-5x \cdot 8x^2 = -40x^3
\][/tex]
- Multiply [tex]\(-5x\)[/tex] by the second term in [tex]\( g(x) \)[/tex], which is [tex]\(-5x\)[/tex]:
[tex]\[
-5x \cdot (-5x) = 25x^2
\][/tex]
- Multiply [tex]\(-5x\)[/tex] by the third term in [tex]\( g(x) \)[/tex], which is [tex]\(-9\)[/tex]:
[tex]\[
-5x \cdot (-9) = 45x
\][/tex]
2. Combine all the terms:
After multiplying, we combine all the results to form a single polynomial:
[tex]\[
(f \cdot g)(x) = -40x^3 + 25x^2 + 45x
\][/tex]
So, the final expression for [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
-40x^3 + 25x^2 + 45x
\][/tex]
This process shows the step-by-step multiplication of the polynomials, resulting in the cubic polynomial given as the solution.
Here's how you would approach it:
1. Distribute [tex]\( f(x) \)[/tex] over each term in [tex]\( g(x) \)[/tex]:
- Multiply [tex]\(-5x\)[/tex] by the first term in [tex]\( g(x) \)[/tex], which is [tex]\( 8x^2 \)[/tex]:
[tex]\[
-5x \cdot 8x^2 = -40x^3
\][/tex]
- Multiply [tex]\(-5x\)[/tex] by the second term in [tex]\( g(x) \)[/tex], which is [tex]\(-5x\)[/tex]:
[tex]\[
-5x \cdot (-5x) = 25x^2
\][/tex]
- Multiply [tex]\(-5x\)[/tex] by the third term in [tex]\( g(x) \)[/tex], which is [tex]\(-9\)[/tex]:
[tex]\[
-5x \cdot (-9) = 45x
\][/tex]
2. Combine all the terms:
After multiplying, we combine all the results to form a single polynomial:
[tex]\[
(f \cdot g)(x) = -40x^3 + 25x^2 + 45x
\][/tex]
So, the final expression for [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
-40x^3 + 25x^2 + 45x
\][/tex]
This process shows the step-by-step multiplication of the polynomials, resulting in the cubic polynomial given as the solution.
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