Answer :
To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] together. Since [tex]\(f(x) = 7x^3 - 5x^2 + 42x - 30\)[/tex] and [tex]\(g(x) = 7x - 5\)[/tex], we will perform polynomial multiplication. Here's a step-by-step explanation:
1. Write down the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 7x^3 - 5x^2 + 42x - 30
\][/tex]
[tex]\[
g(x) = 7x - 5
\][/tex]
2. Multiply each term of [tex]\(f(x)\)[/tex] by each term of [tex]\(g(x)\)[/tex]:
- Multiply [tex]\(7x^3\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
7x^3 \cdot 7x = 49x^4
\][/tex]
[tex]\[
7x^3 \cdot (-5) = -35x^3
\][/tex]
- Multiply [tex]\(-5x^2\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
-5x^2 \cdot 7x = -35x^3
\][/tex]
[tex]\[
-5x^2 \cdot (-5) = 25x^2
\][/tex]
- Multiply [tex]\(42x\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
42x \cdot 7x = 294x^2
\][/tex]
[tex]\[
42x \cdot (-5) = -210x
\][/tex]
- Multiply [tex]\(-30\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
-30 \cdot 7x = -210x
\][/tex]
[tex]\[
-30 \cdot (-5) = 150
\][/tex]
3. Combine all these products:
- [tex]\(49x^4\)[/tex]
- Combine [tex]\(-35x^3\)[/tex] and [tex]\(-35x^3\)[/tex]: [tex]\(-35x^3 - 35x^3 = -70x^3\)[/tex]
- Combine [tex]\(25x^2\)[/tex] and [tex]\(294x^2\)[/tex]: [tex]\(25x^2 + 294x^2 = 319x^2\)[/tex]
- Combine [tex]\(-210x\)[/tex] and [tex]\(-210x\)[/tex]: [tex]\(-210x - 210x = -420x\)[/tex]
- [tex]\(+ 150\)[/tex]
4. Write the final expanded form of [tex]\((f \cdot g)(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150
\][/tex]
This result matches the choice [tex]\((49x^4 - 70x^3 + 319x^2 - 420x + 150)\)[/tex]. So, the correct answer is:
[tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex].
1. Write down the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 7x^3 - 5x^2 + 42x - 30
\][/tex]
[tex]\[
g(x) = 7x - 5
\][/tex]
2. Multiply each term of [tex]\(f(x)\)[/tex] by each term of [tex]\(g(x)\)[/tex]:
- Multiply [tex]\(7x^3\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
7x^3 \cdot 7x = 49x^4
\][/tex]
[tex]\[
7x^3 \cdot (-5) = -35x^3
\][/tex]
- Multiply [tex]\(-5x^2\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
-5x^2 \cdot 7x = -35x^3
\][/tex]
[tex]\[
-5x^2 \cdot (-5) = 25x^2
\][/tex]
- Multiply [tex]\(42x\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
42x \cdot 7x = 294x^2
\][/tex]
[tex]\[
42x \cdot (-5) = -210x
\][/tex]
- Multiply [tex]\(-30\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
[tex]\[
-30 \cdot 7x = -210x
\][/tex]
[tex]\[
-30 \cdot (-5) = 150
\][/tex]
3. Combine all these products:
- [tex]\(49x^4\)[/tex]
- Combine [tex]\(-35x^3\)[/tex] and [tex]\(-35x^3\)[/tex]: [tex]\(-35x^3 - 35x^3 = -70x^3\)[/tex]
- Combine [tex]\(25x^2\)[/tex] and [tex]\(294x^2\)[/tex]: [tex]\(25x^2 + 294x^2 = 319x^2\)[/tex]
- Combine [tex]\(-210x\)[/tex] and [tex]\(-210x\)[/tex]: [tex]\(-210x - 210x = -420x\)[/tex]
- [tex]\(+ 150\)[/tex]
4. Write the final expanded form of [tex]\((f \cdot g)(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150
\][/tex]
This result matches the choice [tex]\((49x^4 - 70x^3 + 319x^2 - 420x + 150)\)[/tex]. So, the correct answer is:
[tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex].
