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Answer :
To multiply the polynomials [tex]\((4x^2 + 3x + 7)(8x - 5)\)[/tex], follow these steps:
1. Distribute each term in the first polynomial to each term in the second polynomial.
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
4x^2 \times 8x = 32x^3
\][/tex]
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
4x^2 \times -5 = -20x^2
\][/tex]
- Multiply [tex]\(3x\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
3x \times 8x = 24x^2
\][/tex]
- Multiply [tex]\(3x\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
3x \times -5 = -15x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
7 \times 8x = 56x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
7 \times -5 = -35
\][/tex]
2. Combine the results by adding like terms together.
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-20x^2 + 24x^2 = 4x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-15x + 56x = 41x
\][/tex]
3. Write the final expression:
After combining all like terms, the expanded polynomial is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
The correct choice for the multiplication of the polynomials is therefore option B: [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
1. Distribute each term in the first polynomial to each term in the second polynomial.
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
4x^2 \times 8x = 32x^3
\][/tex]
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
4x^2 \times -5 = -20x^2
\][/tex]
- Multiply [tex]\(3x\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
3x \times 8x = 24x^2
\][/tex]
- Multiply [tex]\(3x\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
3x \times -5 = -15x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
7 \times 8x = 56x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
7 \times -5 = -35
\][/tex]
2. Combine the results by adding like terms together.
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-20x^2 + 24x^2 = 4x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-15x + 56x = 41x
\][/tex]
3. Write the final expression:
After combining all like terms, the expanded polynomial is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
The correct choice for the multiplication of the polynomials is therefore option B: [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
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