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Answer :
To solve for [tex]\((f \cdot g)(x)\)[/tex] given the functions [tex]\(f(x) = x + 4\)[/tex] and [tex]\(g(x) = 3x^2 - 7\)[/tex], we need to find the product of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
The product [tex]\((f \cdot g)(x)\)[/tex] is calculated as follows:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
Substitute the given functions into the equation:
[tex]\[ (f \cdot g)(x) = (x + 4)(3x^2 - 7) \][/tex]
Next, expand the expression by using the distributive property (i.e., multiply each term in the first polynomial by each term in the second polynomial):
1. Multiply [tex]\(x\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[ x \cdot 3x^2 = 3x^3 \][/tex]
2. Multiply [tex]\(x\)[/tex] by [tex]\(-7\)[/tex]:
[tex]\[ x \cdot (-7) = -7x \][/tex]
3. Multiply [tex]\(4\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[ 4 \cdot 3x^2 = 12x^2 \][/tex]
4. Multiply [tex]\(4\)[/tex] by [tex]\(-7\)[/tex]:
[tex]\[ 4 \cdot (-7) = -28 \][/tex]
Now, combine all these results:
[tex]\[ (f \cdot g)(x) = 3x^3 - 7x + 12x^2 - 28 \][/tex]
Rearrange the terms in descending order of powers of [tex]\(x\)[/tex]:
[tex]\[ (f \cdot g)(x) = 3x^3 + 12x^2 - 7x - 28 \][/tex]
Therefore, the correct answer is:
A. [tex]\((f \cdot g)(x) = 3x^3 + 12x^2 - 7x - 28\)[/tex]
The product [tex]\((f \cdot g)(x)\)[/tex] is calculated as follows:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
Substitute the given functions into the equation:
[tex]\[ (f \cdot g)(x) = (x + 4)(3x^2 - 7) \][/tex]
Next, expand the expression by using the distributive property (i.e., multiply each term in the first polynomial by each term in the second polynomial):
1. Multiply [tex]\(x\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[ x \cdot 3x^2 = 3x^3 \][/tex]
2. Multiply [tex]\(x\)[/tex] by [tex]\(-7\)[/tex]:
[tex]\[ x \cdot (-7) = -7x \][/tex]
3. Multiply [tex]\(4\)[/tex] by [tex]\(3x^2\)[/tex]:
[tex]\[ 4 \cdot 3x^2 = 12x^2 \][/tex]
4. Multiply [tex]\(4\)[/tex] by [tex]\(-7\)[/tex]:
[tex]\[ 4 \cdot (-7) = -28 \][/tex]
Now, combine all these results:
[tex]\[ (f \cdot g)(x) = 3x^3 - 7x + 12x^2 - 28 \][/tex]
Rearrange the terms in descending order of powers of [tex]\(x\)[/tex]:
[tex]\[ (f \cdot g)(x) = 3x^3 + 12x^2 - 7x - 28 \][/tex]
Therefore, the correct answer is:
A. [tex]\((f \cdot g)(x) = 3x^3 + 12x^2 - 7x - 28\)[/tex]
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