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Answer :
To multiply the polynomials [tex]\((7x^2 + 9x + 7)(9x - 4)\)[/tex], we'll distribute each term in the first polynomial by each term in the second polynomial, and then combine like terms. Here's a step-by-step solution:
1. Distribute [tex]\(7x^2\)[/tex] from the first polynomial across each term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[7x^2 \times 9x = 63x^3\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[7x^2 \times -4 = -28x^2\][/tex]
2. Distribute [tex]\(9x\)[/tex] from the first polynomial across each term in the second polynomial:
- Multiply [tex]\(9x\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[9x \times 9x = 81x^2\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[9x \times -4 = -36x\][/tex]
3. Distribute [tex]\(7\)[/tex] from the first polynomial across each term in the second polynomial:
- Multiply [tex]\(7\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[7 \times 9x = 63x\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[7 \times -4 = -28\][/tex]
4. Combine all the terms:
[tex]\[63x^3 + (-28x^2) + 81x^2 + (-36x) + 63x - 28\][/tex]
5. 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]
6. Write the final result:
The product of the polynomials is:
[tex]\[63x^3 + 53x^2 + 27x - 28\][/tex]
Therefore, the correct answer is Option A: [tex]\(63x^3 + 53x^2 + 27x - 28\)[/tex].
1. Distribute [tex]\(7x^2\)[/tex] from the first polynomial across each term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[7x^2 \times 9x = 63x^3\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[7x^2 \times -4 = -28x^2\][/tex]
2. Distribute [tex]\(9x\)[/tex] from the first polynomial across each term in the second polynomial:
- Multiply [tex]\(9x\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[9x \times 9x = 81x^2\][/tex]
- Multiply [tex]\(9x\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[9x \times -4 = -36x\][/tex]
3. Distribute [tex]\(7\)[/tex] from the first polynomial across each term in the second polynomial:
- Multiply [tex]\(7\)[/tex] by [tex]\(9x\)[/tex]:
[tex]\[7 \times 9x = 63x\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-4\)[/tex]:
[tex]\[7 \times -4 = -28\][/tex]
4. Combine all the terms:
[tex]\[63x^3 + (-28x^2) + 81x^2 + (-36x) + 63x - 28\][/tex]
5. 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]
6. Write the final result:
The product of the polynomials is:
[tex]\[63x^3 + 53x^2 + 27x - 28\][/tex]
Therefore, the correct answer is Option A: [tex]\(63x^3 + 53x^2 + 27x - 28\)[/tex].
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