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].
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].
