College

Multiply the polynomials:

[tex]\left(7x^2 + 5x + 7\right)(4x - 6)[/tex]

A. [tex]28x^3 - 62x^2 - 2x - 42[/tex]
B. [tex]28x^3 - 22x^2 - 2x - 42[/tex]
C. [tex]28x^3 - 22x^2 - 58x - 42[/tex]
D. [tex]28x^3 - 22x^2 - 2x + 42[/tex]

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].