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Answer :
To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the two polynomials [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
Given:
[tex]\[ f(x) = 7x^3 - 5x^2 + 42x - 30 \][/tex]
[tex]\[ g(x) = 7x - 5 \][/tex]
We'll multiply each term in [tex]\(f(x)\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
1. Multiply every term in [tex]\(f(x)\)[/tex] by the first term of [tex]\(g(x)\)[/tex], [tex]\(7x\)[/tex]:
- [tex]\(7x \cdot 7x^3 = 49x^4\)[/tex]
- [tex]\(7x \cdot (-5x^2) = -35x^3\)[/tex]
- [tex]\(7x \cdot 42x = 294x^2\)[/tex]
- [tex]\(7x \cdot (-30) = -210x\)[/tex]
2. Multiply every term in [tex]\(f(x)\)[/tex] by the second term of [tex]\(g(x)\)[/tex], [tex]\(-5\)[/tex]:
- [tex]\((-5) \cdot 7x^3 = -35x^3\)[/tex]
- [tex]\((-5) \cdot (-5x^2) = 25x^2\)[/tex]
- [tex]\((-5) \cdot 42x = -210x\)[/tex]
- [tex]\((-5) \cdot (-30) = 150\)[/tex]
Next, add all these results together:
[tex]\[
49x^4 + (-35x^3) + 294x^2 - 210x + (-35x^3) + 25x^2 - 210x + 150
\][/tex]
Now, combine like terms:
- [tex]\(x^4\)[/tex] term: [tex]\(49x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms: [tex]\(-35x^3 + (-35x^3) = -70x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(294x^2 + 25x^2 = 319x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-210x + (-210x) = -420x\)[/tex]
- Constant term: [tex]\(150\)[/tex]
Putting it all together gives the polynomial:
[tex]\[ 49x^4 - 70x^3 + 319x^2 - 420x + 150 \][/tex]
Therefore, [tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex], which matches the first option provided.
Given:
[tex]\[ f(x) = 7x^3 - 5x^2 + 42x - 30 \][/tex]
[tex]\[ g(x) = 7x - 5 \][/tex]
We'll multiply each term in [tex]\(f(x)\)[/tex] by each term in [tex]\(g(x)\)[/tex]:
1. Multiply every term in [tex]\(f(x)\)[/tex] by the first term of [tex]\(g(x)\)[/tex], [tex]\(7x\)[/tex]:
- [tex]\(7x \cdot 7x^3 = 49x^4\)[/tex]
- [tex]\(7x \cdot (-5x^2) = -35x^3\)[/tex]
- [tex]\(7x \cdot 42x = 294x^2\)[/tex]
- [tex]\(7x \cdot (-30) = -210x\)[/tex]
2. Multiply every term in [tex]\(f(x)\)[/tex] by the second term of [tex]\(g(x)\)[/tex], [tex]\(-5\)[/tex]:
- [tex]\((-5) \cdot 7x^3 = -35x^3\)[/tex]
- [tex]\((-5) \cdot (-5x^2) = 25x^2\)[/tex]
- [tex]\((-5) \cdot 42x = -210x\)[/tex]
- [tex]\((-5) \cdot (-30) = 150\)[/tex]
Next, add all these results together:
[tex]\[
49x^4 + (-35x^3) + 294x^2 - 210x + (-35x^3) + 25x^2 - 210x + 150
\][/tex]
Now, combine like terms:
- [tex]\(x^4\)[/tex] term: [tex]\(49x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms: [tex]\(-35x^3 + (-35x^3) = -70x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(294x^2 + 25x^2 = 319x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-210x + (-210x) = -420x\)[/tex]
- Constant term: [tex]\(150\)[/tex]
Putting it all together gives the polynomial:
[tex]\[ 49x^4 - 70x^3 + 319x^2 - 420x + 150 \][/tex]
Therefore, [tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex], which matches the first option provided.
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