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Answer :
To solve the problem of multiplying the polynomials [tex]\((7x^2 + 9x + 7)(9x - 4)\)[/tex], we will apply the distributive property, which involves distributing each term in the first polynomial across each term in the second polynomial.
Here's a detailed, step-by-step solution:
1. Distribute each term in the first polynomial across the second polynomial:
- First, take [tex]\(7x^2\)[/tex] from the first polynomial and multiply it with each term in the second polynomial:
[tex]\[
7x^2 \cdot 9x = 63x^3
\][/tex]
[tex]\[
7x^2 \cdot (-4) = -28x^2
\][/tex]
- Next, take [tex]\(9x\)[/tex] from the first polynomial and multiply it with each term in the second polynomial:
[tex]\[
9x \cdot 9x = 81x^2
\][/tex]
[tex]\[
9x \cdot (-4) = -36x
\][/tex]
- Finally, take [tex]\(7\)[/tex] from the first polynomial and multiply it with each term in the second polynomial:
[tex]\[
7 \cdot 9x = 63x
\][/tex]
[tex]\[
7 \cdot (-4) = -28
\][/tex]
2. Combine all these results:
[tex]\[
63x^3 + (-28x^2) + 81x^2 + (-36x) + 63x + (-28)
\][/tex]
3. Combine like terms:
- For the [tex]\(x^2\)[/tex] terms: [tex]\((-28x^2 + 81x^2 = 53x^2)\)[/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\((-36x + 63x = 27x)\)[/tex]
This gives us:
[tex]\[
63x^3 + 53x^2 + 27x - 28
\][/tex]
The correct answer is therefore:
[tex]\[
\boxed{B. \ 63x^3 + 53x^2 + 27x - 28}
\][/tex]
Here's a detailed, step-by-step solution:
1. Distribute each term in the first polynomial across the second polynomial:
- First, take [tex]\(7x^2\)[/tex] from the first polynomial and multiply it with each term in the second polynomial:
[tex]\[
7x^2 \cdot 9x = 63x^3
\][/tex]
[tex]\[
7x^2 \cdot (-4) = -28x^2
\][/tex]
- Next, take [tex]\(9x\)[/tex] from the first polynomial and multiply it with each term in the second polynomial:
[tex]\[
9x \cdot 9x = 81x^2
\][/tex]
[tex]\[
9x \cdot (-4) = -36x
\][/tex]
- Finally, take [tex]\(7\)[/tex] from the first polynomial and multiply it with each term in the second polynomial:
[tex]\[
7 \cdot 9x = 63x
\][/tex]
[tex]\[
7 \cdot (-4) = -28
\][/tex]
2. Combine all these results:
[tex]\[
63x^3 + (-28x^2) + 81x^2 + (-36x) + 63x + (-28)
\][/tex]
3. Combine like terms:
- For the [tex]\(x^2\)[/tex] terms: [tex]\((-28x^2 + 81x^2 = 53x^2)\)[/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\((-36x + 63x = 27x)\)[/tex]
This gives us:
[tex]\[
63x^3 + 53x^2 + 27x - 28
\][/tex]
The correct answer is therefore:
[tex]\[
\boxed{B. \ 63x^3 + 53x^2 + 27x - 28}
\][/tex]
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