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Answer :
To multiply the polynomials [tex]\((5x^2 + 2x + 8)(7x - 6)\)[/tex], we can use the distributive property. This involves multiplying each term in the first polynomial by each term in the second polynomial, and then combining like terms. Here's how you can do it step-by-step:
1. Distribute each term in [tex]\((5x^2 + 2x + 8)\)[/tex] to each term in [tex]\((7x - 6)\)[/tex]:
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(7x\)[/tex] and [tex]\(5x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x^2 \cdot 7x = 35x^3
\][/tex]
[tex]\[
5x^2 \cdot (-6) = -30x^2
\][/tex]
- Multiply [tex]\(2x\)[/tex] by [tex]\(7x\)[/tex] and [tex]\(2x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
2x \cdot 7x = 14x^2
\][/tex]
[tex]\[
2x \cdot (-6) = -12x
\][/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(7x\)[/tex] and [tex]\(8\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
8 \cdot 7x = 56x
\][/tex]
[tex]\[
8 \cdot (-6) = -48
\][/tex]
2. Combine the results:
Add together all the terms from the products above:
[tex]\[
35x^3 + (-30x^2) + 14x^2 + (-12x) + 56x + (-48)
\][/tex]
3. Combine like terms:
- For [tex]\(x^2\)[/tex] terms:
[tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- For [tex]\(x\)[/tex] terms:
[tex]\(-12x + 56x = 44x\)[/tex]
4. Write the final expression:
Combine the simplified terms:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
Based on these calculations, the correct answer is [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]. So, the answer is C.
1. Distribute each term in [tex]\((5x^2 + 2x + 8)\)[/tex] to each term in [tex]\((7x - 6)\)[/tex]:
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(7x\)[/tex] and [tex]\(5x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x^2 \cdot 7x = 35x^3
\][/tex]
[tex]\[
5x^2 \cdot (-6) = -30x^2
\][/tex]
- Multiply [tex]\(2x\)[/tex] by [tex]\(7x\)[/tex] and [tex]\(2x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
2x \cdot 7x = 14x^2
\][/tex]
[tex]\[
2x \cdot (-6) = -12x
\][/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(7x\)[/tex] and [tex]\(8\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
8 \cdot 7x = 56x
\][/tex]
[tex]\[
8 \cdot (-6) = -48
\][/tex]
2. Combine the results:
Add together all the terms from the products above:
[tex]\[
35x^3 + (-30x^2) + 14x^2 + (-12x) + 56x + (-48)
\][/tex]
3. Combine like terms:
- For [tex]\(x^2\)[/tex] terms:
[tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- For [tex]\(x\)[/tex] terms:
[tex]\(-12x + 56x = 44x\)[/tex]
4. Write the final expression:
Combine the simplified terms:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
Based on these calculations, the correct answer is [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]. So, the answer is C.
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