Answer :
Let's multiply the polynomials [tex]\((4x^2 + 3x + 7)\)[/tex] and [tex]\((8x - 5)\)[/tex] step-by-step.
1. Distribute each term of the first polynomial to every term of the second polynomial:
- Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(8x - 5\)[/tex]:
[tex]\[
4x^2 \times 8x = 32x^3 \\
4x^2 \times -5 = -20x^2
\][/tex]
- Multiply [tex]\(3x\)[/tex] by each term in [tex]\(8x - 5\)[/tex]:
[tex]\[
3x \times 8x = 24x^2 \\
3x \times -5 = -15x
\][/tex]
- Multiply [tex]\(7\)[/tex] by each term in [tex]\(8x - 5\)[/tex]:
[tex]\[
7 \times 8x = 56x \\
7 \times -5 = -35
\][/tex]
2. Combine all these results:
[tex]\[
32x^3 + (-20x^2 + 24x^2) + (-15x + 56x) + (-35)
\][/tex]
3. Simplify by combining like terms:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Now, let's compare this result with the given options:
- A. [tex]\(32x^3 - 44x^2 - 71x - 35\)[/tex]
- B. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex]
- C. [tex]\(32x^3 + 4x^2 + 41x + 35\)[/tex]
- D. [tex]\(32x^3 - 4x^2 - 41x + 35\)[/tex]
The correct answer is B. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
1. Distribute each term of the first polynomial to every term of the second polynomial:
- Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(8x - 5\)[/tex]:
[tex]\[
4x^2 \times 8x = 32x^3 \\
4x^2 \times -5 = -20x^2
\][/tex]
- Multiply [tex]\(3x\)[/tex] by each term in [tex]\(8x - 5\)[/tex]:
[tex]\[
3x \times 8x = 24x^2 \\
3x \times -5 = -15x
\][/tex]
- Multiply [tex]\(7\)[/tex] by each term in [tex]\(8x - 5\)[/tex]:
[tex]\[
7 \times 8x = 56x \\
7 \times -5 = -35
\][/tex]
2. Combine all these results:
[tex]\[
32x^3 + (-20x^2 + 24x^2) + (-15x + 56x) + (-35)
\][/tex]
3. Simplify by combining like terms:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Now, let's compare this result with the given options:
- A. [tex]\(32x^3 - 44x^2 - 71x - 35\)[/tex]
- B. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex]
- C. [tex]\(32x^3 + 4x^2 + 41x + 35\)[/tex]
- D. [tex]\(32x^3 - 4x^2 - 41x + 35\)[/tex]
The correct answer is B. [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
