Answer :
To find [tex]\((f \cdot g)(x)\)[/tex] for the functions [tex]\(f(x) = 7x^3 - 5x^2 + 42x - 30\)[/tex] and [tex]\(g(x) = 7x - 5\)[/tex], we need to multiply these two functions. This involves distributing each term in [tex]\(f(x)\)[/tex] to each term in [tex]\(g(x)\)[/tex], and then combining like terms. Here is a step-by-step guide:
1. Identify the Terms:
- [tex]\(f(x)\)[/tex] has terms: [tex]\(7x^3, -5x^2, 42x, -30\)[/tex].
- [tex]\(g(x)\)[/tex] has terms: [tex]\(7x, -5\)[/tex].
2. Distribute each term of [tex]\(f(x)\)[/tex] to 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 the terms:
Collect all the results from the distribution:
[tex]\[
49x^4 + (-35x^3 - 35x^3) + (25x^2 + 294x^2) + (-210x - 210x) + 150
\][/tex]
4. Simplify the expression:
- Combine like terms:
- [tex]\(x^3\)[/tex] terms: [tex]\(-35x^3 - 35x^3 = -70x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(25x^2 + 294x^2 = 319x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-210x - 210x = -420x\)[/tex]
5. Final expression:
[tex]\[
(f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150
\][/tex]
Thus, the polynomial representing [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
\boxed{49x^4 - 70x^3 + 319x^2 - 420x + 150}
\][/tex]
1. Identify the Terms:
- [tex]\(f(x)\)[/tex] has terms: [tex]\(7x^3, -5x^2, 42x, -30\)[/tex].
- [tex]\(g(x)\)[/tex] has terms: [tex]\(7x, -5\)[/tex].
2. Distribute each term of [tex]\(f(x)\)[/tex] to 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 the terms:
Collect all the results from the distribution:
[tex]\[
49x^4 + (-35x^3 - 35x^3) + (25x^2 + 294x^2) + (-210x - 210x) + 150
\][/tex]
4. Simplify the expression:
- Combine like terms:
- [tex]\(x^3\)[/tex] terms: [tex]\(-35x^3 - 35x^3 = -70x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(25x^2 + 294x^2 = 319x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-210x - 210x = -420x\)[/tex]
5. Final expression:
[tex]\[
(f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150
\][/tex]
Thus, the polynomial representing [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\[
\boxed{49x^4 - 70x^3 + 319x^2 - 420x + 150}
\][/tex]
