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Answer :
To multiply the polynomials [tex]\((7x^2 + 9x + 7)(9x - 4)\)[/tex], we will use the distributive property (also known as the FOIL method for binomials), which involves multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms.
Here’s the step-by-step process:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- First, multiply [tex]\(7x^2\)[/tex] by both terms 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 both terms in [tex]\(9x - 4\)[/tex]:
- [tex]\(9x \times 9x = 81x^2\)[/tex]
- [tex]\(9x \times (-4) = -36x\)[/tex]
- Finally, multiply [tex]\(7\)[/tex] by both terms in [tex]\(9x - 4\)[/tex]:
- [tex]\(7 \times 9x = 63x\)[/tex]
- [tex]\(7 \times (-4) = -28\)[/tex]
2. Combine all the products:
[tex]\[
63x^3 + (-28x^2) + 81x^2 + (-36x) + 63x - 28
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-28x^2 + 81x^2 = 53x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-36x + 63x = 27x\)[/tex]
4. Write the final result:
[tex]\[
63x^3 + 53x^2 + 27x - 28
\][/tex]
The correct answer, therefore, is:
A. [tex]\(63x^3 + 53x^2 + 27x - 28\)[/tex]
Here’s the step-by-step process:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- First, multiply [tex]\(7x^2\)[/tex] by both terms 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 both terms in [tex]\(9x - 4\)[/tex]:
- [tex]\(9x \times 9x = 81x^2\)[/tex]
- [tex]\(9x \times (-4) = -36x\)[/tex]
- Finally, multiply [tex]\(7\)[/tex] by both terms in [tex]\(9x - 4\)[/tex]:
- [tex]\(7 \times 9x = 63x\)[/tex]
- [tex]\(7 \times (-4) = -28\)[/tex]
2. Combine all the products:
[tex]\[
63x^3 + (-28x^2) + 81x^2 + (-36x) + 63x - 28
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-28x^2 + 81x^2 = 53x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-36x + 63x = 27x\)[/tex]
4. Write the final result:
[tex]\[
63x^3 + 53x^2 + 27x - 28
\][/tex]
The correct answer, therefore, is:
A. [tex]\(63x^3 + 53x^2 + 27x - 28\)[/tex]
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