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Answer :
Sure! Let's multiply the polynomials [tex]\((7x^2 + 5x + 7)\)[/tex] and [tex]\((4x - 6)\)[/tex] step-by-step:
1. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by each term in the second polynomial:
- [tex]\(7x^2 \times 4x = 28x^3\)[/tex]
- [tex]\(7x^2 \times -6 = -42x^2\)[/tex]
- Multiply [tex]\(5x\)[/tex] by each term in the second polynomial:
- [tex]\(5x \times 4x = 20x^2\)[/tex]
- [tex]\(5x \times -6 = -30x\)[/tex]
- Multiply [tex]\(7\)[/tex] by each term in the second polynomial:
- [tex]\(7 \times 4x = 28x\)[/tex]
- [tex]\(7 \times -6 = -42\)[/tex]
2. Combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\(28x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex]
- The constant term: [tex]\(-42\)[/tex]
3. Write the combined expression:
The result of multiplying the two polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
So, the correct answer is option B: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
1. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by each term in the second polynomial:
- [tex]\(7x^2 \times 4x = 28x^3\)[/tex]
- [tex]\(7x^2 \times -6 = -42x^2\)[/tex]
- Multiply [tex]\(5x\)[/tex] by each term in the second polynomial:
- [tex]\(5x \times 4x = 20x^2\)[/tex]
- [tex]\(5x \times -6 = -30x\)[/tex]
- Multiply [tex]\(7\)[/tex] by each term in the second polynomial:
- [tex]\(7 \times 4x = 28x\)[/tex]
- [tex]\(7 \times -6 = -42\)[/tex]
2. Combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\(28x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex]
- The constant term: [tex]\(-42\)[/tex]
3. Write the combined expression:
The result of multiplying the two polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
So, the correct answer is option B: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
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