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Answer :
To multiply the polynomials [tex]\((4x^2 + 3x + 7)\)[/tex] and [tex]\((8x - 5)\)[/tex], we'll go through the multiplication process step by step. Here's how you do it:
1. Multiply each term of the first polynomial by each term of the second polynomial.
- Start by distributing [tex]\(4x^2\)[/tex] across each term in [tex]\((8x - 5)\)[/tex]:
[tex]\[
4x^2 \cdot 8x = 32x^3
\][/tex]
[tex]\[
4x^2 \cdot (-5) = -20x^2
\][/tex]
- Next, distribute [tex]\(3x\)[/tex] across each term in [tex]\((8x - 5)\)[/tex]:
[tex]\[
3x \cdot 8x = 24x^2
\][/tex]
[tex]\[
3x \cdot (-5) = -15x
\][/tex]
- Finally, distribute [tex]\(7\)[/tex] across each term in [tex]\((8x - 5)\)[/tex]:
[tex]\[
7 \cdot 8x = 56x
\][/tex]
[tex]\[
7 \cdot (-5) = -35
\][/tex]
2. Combine all the terms:
Now, let's collect all the resulting terms together:
- [tex]\(32x^3\)[/tex]
- 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]
- [tex]\(-35\)[/tex] is the constant term.
3. Write the final polynomial:
Thus, the product of the polynomials is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
This matches option D: [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
1. Multiply each term of the first polynomial by each term of the second polynomial.
- Start by distributing [tex]\(4x^2\)[/tex] across each term in [tex]\((8x - 5)\)[/tex]:
[tex]\[
4x^2 \cdot 8x = 32x^3
\][/tex]
[tex]\[
4x^2 \cdot (-5) = -20x^2
\][/tex]
- Next, distribute [tex]\(3x\)[/tex] across each term in [tex]\((8x - 5)\)[/tex]:
[tex]\[
3x \cdot 8x = 24x^2
\][/tex]
[tex]\[
3x \cdot (-5) = -15x
\][/tex]
- Finally, distribute [tex]\(7\)[/tex] across each term in [tex]\((8x - 5)\)[/tex]:
[tex]\[
7 \cdot 8x = 56x
\][/tex]
[tex]\[
7 \cdot (-5) = -35
\][/tex]
2. Combine all the terms:
Now, let's collect all the resulting terms together:
- [tex]\(32x^3\)[/tex]
- 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]
- [tex]\(-35\)[/tex] is the constant term.
3. Write the final polynomial:
Thus, the product of the polynomials is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
This matches option D: [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
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