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Answer :
To multiply the polynomials [tex]\((4x^2 + 3x + 7)(8x - 5)\)[/tex], we'll use the distributive property, also known as the FOIL method when applied to binomials, to ensure each term from the first polynomial is multiplied with each term from the second polynomial.
Here's a step-by-step approach:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- First, multiply [tex]\(4x^2\)[/tex] by both terms in [tex]\(8x - 5\)[/tex]:
[tex]\[
4x^2 \cdot 8x = 32x^3
\][/tex]
[tex]\[
4x^2 \cdot (-5) = -20x^2
\][/tex]
- Next, multiply [tex]\(3x\)[/tex] by both terms in [tex]\(8x - 5\)[/tex]:
[tex]\[
3x \cdot 8x = 24x^2
\][/tex]
[tex]\[
3x \cdot (-5) = -15x
\][/tex]
- Finally, multiply [tex]\(7\)[/tex] by both terms in [tex]\(8x - 5\)[/tex]:
[tex]\[
7 \cdot 8x = 56x
\][/tex]
[tex]\[
7 \cdot (-5) = -35
\][/tex]
2. Combine all the results from the multiplications:
The products obtained from each term are:
[tex]\[
32x^3, \, -20x^2, \, 24x^2, \, -15x, \, 56x, \, -35
\][/tex]
3. Combine like terms:
- 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]
4. Write the final expression:
The complete expanded expression, combining all terms, is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Thus, the correct answer is [tex]\(\boxed{32x^3 + 4x^2 + 41x - 35}\)[/tex], which corresponds to option A.
Here's a step-by-step approach:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- First, multiply [tex]\(4x^2\)[/tex] by both terms in [tex]\(8x - 5\)[/tex]:
[tex]\[
4x^2 \cdot 8x = 32x^3
\][/tex]
[tex]\[
4x^2 \cdot (-5) = -20x^2
\][/tex]
- Next, multiply [tex]\(3x\)[/tex] by both terms in [tex]\(8x - 5\)[/tex]:
[tex]\[
3x \cdot 8x = 24x^2
\][/tex]
[tex]\[
3x \cdot (-5) = -15x
\][/tex]
- Finally, multiply [tex]\(7\)[/tex] by both terms in [tex]\(8x - 5\)[/tex]:
[tex]\[
7 \cdot 8x = 56x
\][/tex]
[tex]\[
7 \cdot (-5) = -35
\][/tex]
2. Combine all the results from the multiplications:
The products obtained from each term are:
[tex]\[
32x^3, \, -20x^2, \, 24x^2, \, -15x, \, 56x, \, -35
\][/tex]
3. Combine like terms:
- 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]
4. Write the final expression:
The complete expanded expression, combining all terms, is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Thus, the correct answer is [tex]\(\boxed{32x^3 + 4x^2 + 41x - 35}\)[/tex], which corresponds to option A.
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