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Answer :
To multiply the polynomials [tex]\((7x^2 + 5x + 7)(4x - 6)\)[/tex], we will use the distributive property. We will multiply each term in the first polynomial by each term in the second polynomial and then combine the like terms.
1. 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]
2. Multiply [tex]\(5x\)[/tex] by each term in the second polynomial:
- [tex]\(5x \times 4x = 20x^2\)[/tex]
- [tex]\(5x \times (-6) = -30x\)[/tex]
3. Multiply [tex]\(7\)[/tex] by each term in the second polynomial:
- [tex]\(7 \times 4x = 28x\)[/tex]
- [tex]\(7 \times (-6) = -42\)[/tex]
Now, add all these results together:
- 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] term: [tex]\(-30x + 28x = -2x\)[/tex]
- The constant term: [tex]\(-42\)[/tex]
Therefore, the product of the polynomials is:
[tex]\[ 28x^3 - 22x^2 - 2x - 42 \][/tex]
This matches the option:
B. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex]
1. 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]
2. Multiply [tex]\(5x\)[/tex] by each term in the second polynomial:
- [tex]\(5x \times 4x = 20x^2\)[/tex]
- [tex]\(5x \times (-6) = -30x\)[/tex]
3. Multiply [tex]\(7\)[/tex] by each term in the second polynomial:
- [tex]\(7 \times 4x = 28x\)[/tex]
- [tex]\(7 \times (-6) = -42\)[/tex]
Now, add all these results together:
- 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] term: [tex]\(-30x + 28x = -2x\)[/tex]
- The constant term: [tex]\(-42\)[/tex]
Therefore, the product of the polynomials is:
[tex]\[ 28x^3 - 22x^2 - 2x - 42 \][/tex]
This matches the option:
B. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex]
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