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Answer :
Let's multiply the polynomials [tex]\((7x^2 + 5x + 7)(4x - 6)\)[/tex] step-by-step to find the result.
1. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply the first term in the first polynomial by each term in the second polynomial:
[tex]\[
7x^2 \cdot 4x = 28x^3
\][/tex]
[tex]\[
7x^2 \cdot (-6) = -42x^2
\][/tex]
- Multiply the second term in the first polynomial by each term in the second polynomial:
[tex]\[
5x \cdot 4x = 20x^2
\][/tex]
[tex]\[
5x \cdot (-6) = -30x
\][/tex]
- Multiply the third term in the first polynomial by each term in the second polynomial:
[tex]\[
7 \cdot 4x = 28x
\][/tex]
[tex]\[
7 \cdot (-6) = -42
\][/tex]
2. Combine the like terms from all the terms obtained:
- Combine [tex]\(x^3\)[/tex] terms:
[tex]\[
28x^3
\][/tex]
- Combine [tex]\(x^2\)[/tex] terms:
[tex]\[
-42x^2 + 20x^2 = -22x^2
\][/tex]
- Combine [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 28x = -2x
\][/tex]
- The constant term is:
[tex]\[
-42
\][/tex]
Thus, the result of multiplying the polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
Therefore, the correct answer is C. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
1. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply the first term in the first polynomial by each term in the second polynomial:
[tex]\[
7x^2 \cdot 4x = 28x^3
\][/tex]
[tex]\[
7x^2 \cdot (-6) = -42x^2
\][/tex]
- Multiply the second term in the first polynomial by each term in the second polynomial:
[tex]\[
5x \cdot 4x = 20x^2
\][/tex]
[tex]\[
5x \cdot (-6) = -30x
\][/tex]
- Multiply the third term in the first polynomial by each term in the second polynomial:
[tex]\[
7 \cdot 4x = 28x
\][/tex]
[tex]\[
7 \cdot (-6) = -42
\][/tex]
2. Combine the like terms from all the terms obtained:
- Combine [tex]\(x^3\)[/tex] terms:
[tex]\[
28x^3
\][/tex]
- Combine [tex]\(x^2\)[/tex] terms:
[tex]\[
-42x^2 + 20x^2 = -22x^2
\][/tex]
- Combine [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 28x = -2x
\][/tex]
- The constant term is:
[tex]\[
-42
\][/tex]
Thus, the result of multiplying the polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
Therefore, the correct answer is C. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
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