College

Find \((f \cdot g)(x)\) if \[f(x) = 7x^3 - 5x^2 + 42x - 30\] and \[g(x) = 7x - 5\].

A. \((f \cdot g)(x) = 49x^4 - 269x^2 - 150\)

B. \((f \cdot g)(x) = 49x^4 + 269x^2 + 150\)

C. \((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)

D. \((f \cdot g)(x) = 49x^4 + 70x^3 - 319x^2 + 420x - 150\)

Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. The given functions are:

[tex]\[ f(x) = 7x^3 - 5x^2 + 42x - 30 \][/tex]

[tex]\[ g(x) = 7x - 5 \][/tex]

To perform the multiplication, we distribute each term in [tex]\(f(x)\)[/tex] with each term in [tex]\(g(x)\)[/tex]:

1. 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]

2. 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]

3. 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]

4. Multiply [tex]\(-30\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
- [tex]\(-30 \cdot 7x = -210x\)[/tex]
- [tex]\(-30 \cdot (-5) = 150\)[/tex]

Now, combine all these terms together:

[tex]\[
(f \cdot g)(x) = 49x^4 - 35x^3 - 35x^3 + 25x^2 + 294x^2 - 210x - 210x + 150
\][/tex]

Next, combine like terms:

- The [tex]\(x^4\)[/tex] term: [tex]\(49x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-35x^3 - 35x^3 = -70x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(25x^2 + 294x^2 = 319x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-210x - 210x = -420x\)[/tex]
- The constant term: [tex]\(150\)[/tex]

The resulting expression is:

[tex]\[
(f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150
\][/tex]

Therefore, the answer is:

[tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex].

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