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Answer :
To multiply the polynomials [tex]\((7x^2 + 9x + 7)(9x - 4)\)[/tex], we need to apply the distributive property, often referred to as the FOIL method when dealing with binomials, although in this case, we are working with a trinomial and a binomial. Here's how you can solve it step-by-step:
1. Distribute each term in the first polynomial to each term in the second polynomial:
- First, multiply [tex]\(7x^2\)[/tex] by each term in [tex]\(9x - 4\)[/tex]:
[tex]\[
7x^2 \times 9x = 63x^3
\][/tex]
[tex]\[
7x^2 \times (-4) = -28x^2
\][/tex]
- Next, multiply [tex]\(9x\)[/tex] by each term in [tex]\(9x - 4\)[/tex]:
[tex]\[
9x \times 9x = 81x^2
\][/tex]
[tex]\[
9x \times (-4) = -36x
\][/tex]
- Finally, multiply [tex]\(7\)[/tex] by each term in [tex]\(9x - 4\)[/tex]:
[tex]\[
7 \times 9x = 63x
\][/tex]
[tex]\[
7 \times (-4) = -28
\][/tex]
2. Combine all these products:
After distributing, you will have:
[tex]\[
63x^3 + (-28x^2) + 81x^2 + (-36x) + 63x + (-28)
\][/tex]
3. Combine like terms:
- For [tex]\(x^2\)[/tex] terms: [tex]\(-28x^2 + 81x^2 = 53x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(-36x + 63x = 27x\)[/tex]
4. Write the simplified polynomial:
Combine all the terms:
[tex]\[
63x^3 + 53x^2 + 27x - 28
\][/tex]
Therefore, the answer is D. [tex]\(63x^3 + 53x^2 + 27x - 28\)[/tex].
1. Distribute each term in the first polynomial to each term in the second polynomial:
- First, multiply [tex]\(7x^2\)[/tex] by each term in [tex]\(9x - 4\)[/tex]:
[tex]\[
7x^2 \times 9x = 63x^3
\][/tex]
[tex]\[
7x^2 \times (-4) = -28x^2
\][/tex]
- Next, multiply [tex]\(9x\)[/tex] by each term in [tex]\(9x - 4\)[/tex]:
[tex]\[
9x \times 9x = 81x^2
\][/tex]
[tex]\[
9x \times (-4) = -36x
\][/tex]
- Finally, multiply [tex]\(7\)[/tex] by each term in [tex]\(9x - 4\)[/tex]:
[tex]\[
7 \times 9x = 63x
\][/tex]
[tex]\[
7 \times (-4) = -28
\][/tex]
2. Combine all these products:
After distributing, you will have:
[tex]\[
63x^3 + (-28x^2) + 81x^2 + (-36x) + 63x + (-28)
\][/tex]
3. Combine like terms:
- For [tex]\(x^2\)[/tex] terms: [tex]\(-28x^2 + 81x^2 = 53x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(-36x + 63x = 27x\)[/tex]
4. Write the simplified polynomial:
Combine all the terms:
[tex]\[
63x^3 + 53x^2 + 27x - 28
\][/tex]
Therefore, the answer is D. [tex]\(63x^3 + 53x^2 + 27x - 28\)[/tex].
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