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Answer :
Sure! Let's multiply the polynomials step by step.
We need to multiply the polynomials [tex]\((4x^2 + 3x + 7)\)[/tex] and [tex]\((8x - 5)\)[/tex].
Here's the step-by-step process:
1. Distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[
(4x^2 + 3x + 7)(8x - 5)
\][/tex]
2. First, distribute [tex]\(4x^2\)[/tex] across [tex]\(8x - 5\)[/tex]:
[tex]\[
4x^2 \cdot (8x - 5) = 4x^2 \cdot 8x + 4x^2 \cdot (-5) = 32x^3 - 20x^2
\][/tex]
3. Next, distribute [tex]\(3x\)[/tex] across [tex]\(8x - 5\)[/tex]:
[tex]\[
3x \cdot (8x - 5) = 3x \cdot 8x + 3x \cdot (-5) = 24x^2 - 15x
\][/tex]
4. Finally, distribute [tex]\(7\)[/tex] across [tex]\(8x - 5\)[/tex]:
[tex]\[
7 \cdot (8x - 5) = 7 \cdot 8x + 7 \cdot (-5) = 56x - 35
\][/tex]
5. Combine all the distributed parts together:
[tex]\[
32x^3 - 20x^2 + 24x^2 - 15x + 56x - 35
\][/tex]
6. Combine like terms:
- For [tex]\(x^2\)[/tex]: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-15x + 56x = 41x\)[/tex]
So, we have:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{32x^3 + 4x^2 + 41x - 35}
\][/tex]
Looking at the options provided:
A. [tex]\(32x^3 - 4x^2 - 41x + 35\)[/tex]
B. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex]
C. [tex]\(32x^3 + 4x^2 + 41x + 35\)[/tex]
D. [tex]\(32x^3 - 44x^2 - 71x - 35\)[/tex]
The correct choice is [tex]\(B\)[/tex]:
[tex]\[
\boxed{32x^3 + 4x^2 + 41x - 35}
\][/tex]
We need to multiply the polynomials [tex]\((4x^2 + 3x + 7)\)[/tex] and [tex]\((8x - 5)\)[/tex].
Here's the step-by-step process:
1. Distribute each term in the first polynomial by each term in the second polynomial:
[tex]\[
(4x^2 + 3x + 7)(8x - 5)
\][/tex]
2. First, distribute [tex]\(4x^2\)[/tex] across [tex]\(8x - 5\)[/tex]:
[tex]\[
4x^2 \cdot (8x - 5) = 4x^2 \cdot 8x + 4x^2 \cdot (-5) = 32x^3 - 20x^2
\][/tex]
3. Next, distribute [tex]\(3x\)[/tex] across [tex]\(8x - 5\)[/tex]:
[tex]\[
3x \cdot (8x - 5) = 3x \cdot 8x + 3x \cdot (-5) = 24x^2 - 15x
\][/tex]
4. Finally, distribute [tex]\(7\)[/tex] across [tex]\(8x - 5\)[/tex]:
[tex]\[
7 \cdot (8x - 5) = 7 \cdot 8x + 7 \cdot (-5) = 56x - 35
\][/tex]
5. Combine all the distributed parts together:
[tex]\[
32x^3 - 20x^2 + 24x^2 - 15x + 56x - 35
\][/tex]
6. Combine like terms:
- For [tex]\(x^2\)[/tex]: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-15x + 56x = 41x\)[/tex]
So, we have:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{32x^3 + 4x^2 + 41x - 35}
\][/tex]
Looking at the options provided:
A. [tex]\(32x^3 - 4x^2 - 41x + 35\)[/tex]
B. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex]
C. [tex]\(32x^3 + 4x^2 + 41x + 35\)[/tex]
D. [tex]\(32x^3 - 44x^2 - 71x - 35\)[/tex]
The correct choice is [tex]\(B\)[/tex]:
[tex]\[
\boxed{32x^3 + 4x^2 + 41x - 35}
\][/tex]
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