Answer :
To multiply the polynomials [tex]\((7x^2 + 5x + 7)\)[/tex] and [tex]\((4x - 6)\)[/tex], we'll use the distributive property to ensure each term in the first polynomial is multiplied by each term in the second polynomial.
Here's how you can perform the multiplication step-by-step:
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]
4. Combine all these results:
- Combine all the [tex]\(x^3\)[/tex] terms: [tex]\(28x^3\)[/tex]
- Combine all the [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex]
- Combine all the [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex]
- Combine the constant terms: [tex]\(-42\)[/tex]
5. Write out the simplified polynomial:
The result of multiplying the two polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
Thus, the solution is option B: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
Here's how you can perform the multiplication step-by-step:
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]
4. Combine all these results:
- Combine all the [tex]\(x^3\)[/tex] terms: [tex]\(28x^3\)[/tex]
- Combine all the [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex]
- Combine all the [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex]
- Combine the constant terms: [tex]\(-42\)[/tex]
5. Write out the simplified polynomial:
The result of multiplying the two polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
Thus, the solution is option B: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
