College

What is the product of the polynomials below?

[tex]\left(5x^2 - x - 3\right)(2x + 6)[/tex]

A. [tex]10x^3 + 28x^2 + 12x + 3[/tex]
B. [tex]10x^3 + 28x^2 - 12x - 18[/tex]
C. [tex]10x^3 + 28x^2 + 12x + 18[/tex]
D. [tex]10x^3 + 28x^2 - 12x - 3[/tex]

Answer :

To find the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex], we'll multiply each term in the first polynomial by each term in the second polynomial using the distributive property. Here's the step-by-step breakdown:

1. Distribute [tex]\(5x^2\)[/tex] from the first polynomial:
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(10x^3\)[/tex].
- Multiply [tex]\(5x^2\)[/tex] by [tex]\(6\)[/tex] to get [tex]\(30x^2\)[/tex].

2. Distribute [tex]\(-x\)[/tex] from the first polynomial:
- Multiply [tex]\(-x\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(-2x^2\)[/tex].
- Multiply [tex]\(-x\)[/tex] by [tex]\(6\)[/tex] to get [tex]\(-6x\)[/tex].

3. Distribute [tex]\(-3\)[/tex] from the first polynomial:
- Multiply [tex]\(-3\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(-6x\)[/tex].
- Multiply [tex]\(-3\)[/tex] by [tex]\(6\)[/tex] to get [tex]\(-18\)[/tex].

4. Combine all the results:
- Collecting all products together:
[tex]\(10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18\)[/tex].

5. Simplify the expression:
- Combine like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(30x^2 - 2x^2 = 28x^2\)[/tex].
- The [tex]\(x\)[/tex] terms: [tex]\(-6x - 6x = -12x\)[/tex].
- So the expression becomes:
[tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].

The correct answer is:
B. [tex]\(10x^3 + 28x^2 - 12x - 18\)[/tex].