Answer :
To find the product of the polynomials [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex], we can use the distributive property to expand the expression. Here's how to do it step by step:
1. Distribute each term of the first polynomial to each term of the second polynomial.
- First, take [tex]\(2x^2\)[/tex] and multiply it by each term in the second polynomial:
[tex]\[
2x^2 \cdot 3x = 6x^3
\][/tex]
[tex]\[
2x^2 \cdot 5 = 10x^2
\][/tex]
- Next, take [tex]\(3x\)[/tex] and multiply it by each term in the second polynomial:
[tex]\[
3x \cdot 3x = 9x^2
\][/tex]
[tex]\[
3x \cdot 5 = 15x
\][/tex]
- Finally, take [tex]\(-1\)[/tex] and multiply it by each term in the second polynomial:
[tex]\[
-1 \cdot 3x = -3x
\][/tex]
[tex]\[
-1 \cdot 5 = -5
\][/tex]
2. Combine all these results together.
[tex]\[
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5
\][/tex]
3. Combine like terms.
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 + 9x^2 = 19x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(15x - 3x = 12x\)[/tex]
4. Write the final expression:
[tex]\[
6x^3 + 19x^2 + 12x - 5
\][/tex]
So, the product of [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex] is [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
The correct answer is D. [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
1. Distribute each term of the first polynomial to each term of the second polynomial.
- First, take [tex]\(2x^2\)[/tex] and multiply it by each term in the second polynomial:
[tex]\[
2x^2 \cdot 3x = 6x^3
\][/tex]
[tex]\[
2x^2 \cdot 5 = 10x^2
\][/tex]
- Next, take [tex]\(3x\)[/tex] and multiply it by each term in the second polynomial:
[tex]\[
3x \cdot 3x = 9x^2
\][/tex]
[tex]\[
3x \cdot 5 = 15x
\][/tex]
- Finally, take [tex]\(-1\)[/tex] and multiply it by each term in the second polynomial:
[tex]\[
-1 \cdot 3x = -3x
\][/tex]
[tex]\[
-1 \cdot 5 = -5
\][/tex]
2. Combine all these results together.
[tex]\[
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5
\][/tex]
3. Combine like terms.
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 + 9x^2 = 19x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(15x - 3x = 12x\)[/tex]
4. Write the final expression:
[tex]\[
6x^3 + 19x^2 + 12x - 5
\][/tex]
So, the product of [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex] is [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
The correct answer is D. [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
