Answer :
Let's multiply the polynomials [tex]\((7x^2 + 5x + 7)(4x - 6)\)[/tex] together, step by step.
1. Distribute each term in the first polynomial to every term in the second polynomial:
- Start with [tex]\(7x^2 \times (4x - 6)\)[/tex]:
- [tex]\(7x^2 \times 4x = 28x^3\)[/tex]
- [tex]\(7x^2 \times -6 = -42x^2\)[/tex]
- Next, distribute [tex]\(5x\)[/tex] across [tex]\(4x - 6\)[/tex]:
- [tex]\(5x \times 4x = 20x^2\)[/tex]
- [tex]\(5x \times -6 = -30x\)[/tex]
- Lastly, distribute [tex]\(7\)[/tex] across [tex]\(4x - 6\)[/tex]:
- [tex]\(7 \times 4x = 28x\)[/tex]
- [tex]\(7 \times -6 = -42\)[/tex]
2. Combine all the terms:
The expression becomes:
[tex]\(28x^3 + (-42x^2) + 20x^2 + (-30x) + 28x + (-42)\)[/tex].
3. Combine like terms:
- For [tex]\(x^3\)[/tex], we only have one term: [tex]\(28x^3\)[/tex].
- For [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex].
- For [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex].
- The constant term is [tex]\(-42\)[/tex].
Putting it all together, the result of multiplying the polynomials is:
[tex]\[ 28x^3 - 22x^2 - 2x - 42 \][/tex]
Therefore, the correct answer is C. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
1. Distribute each term in the first polynomial to every term in the second polynomial:
- Start with [tex]\(7x^2 \times (4x - 6)\)[/tex]:
- [tex]\(7x^2 \times 4x = 28x^3\)[/tex]
- [tex]\(7x^2 \times -6 = -42x^2\)[/tex]
- Next, distribute [tex]\(5x\)[/tex] across [tex]\(4x - 6\)[/tex]:
- [tex]\(5x \times 4x = 20x^2\)[/tex]
- [tex]\(5x \times -6 = -30x\)[/tex]
- Lastly, distribute [tex]\(7\)[/tex] across [tex]\(4x - 6\)[/tex]:
- [tex]\(7 \times 4x = 28x\)[/tex]
- [tex]\(7 \times -6 = -42\)[/tex]
2. Combine all the terms:
The expression becomes:
[tex]\(28x^3 + (-42x^2) + 20x^2 + (-30x) + 28x + (-42)\)[/tex].
3. Combine like terms:
- For [tex]\(x^3\)[/tex], we only have one term: [tex]\(28x^3\)[/tex].
- For [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex].
- For [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex].
- The constant term is [tex]\(-42\)[/tex].
Putting it all together, the result of multiplying the polynomials is:
[tex]\[ 28x^3 - 22x^2 - 2x - 42 \][/tex]
Therefore, the correct answer is C. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
