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Answer :
Sure, let's find the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex] step-by-step.
1. Distribute Each Term in the First Polynomial to the Second Polynomial:
To do this, distribute each term in the first polynomial, [tex]\(5x^2 - x - 3\)[/tex], to each term in the second polynomial, [tex]\(2x + 6\)[/tex].
- Distribute [tex]\(5x^2\)[/tex] to each term in [tex]\(2x + 6\)[/tex]:
[tex]\[
5x^2 \cdot 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \cdot 6 = 30x^2
\][/tex]
- Distribute [tex]\(-x\)[/tex] to each term in [tex]\(2x + 6\)[/tex]:
[tex]\[
-x \cdot 2x = -2x^2
\][/tex]
[tex]\[
-x \cdot 6 = -6x
\][/tex]
- Distribute [tex]\(-3\)[/tex] to each term in [tex]\(2x + 6\)[/tex]:
[tex]\[
-3 \cdot 2x = -6x
\][/tex]
[tex]\[
-3 \cdot 6 = -18
\][/tex]
2. Combine All These Results:
Now, add all the products together:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
3. Combine Like Terms:
Combine the like terms ([tex]\(30x^2 - 2x^2\)[/tex] and [tex]\(-6x - 6x\)[/tex]):
[tex]\[
10x^3 + (30x^2 - 2x^2) + (-6x - 6x) - 18
\][/tex]
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So, the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex] is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
The correct answer is:
C. [tex]\(10 x^3 + 28 x^2 - 12 x - 18\)[/tex]
1. Distribute Each Term in the First Polynomial to the Second Polynomial:
To do this, distribute each term in the first polynomial, [tex]\(5x^2 - x - 3\)[/tex], to each term in the second polynomial, [tex]\(2x + 6\)[/tex].
- Distribute [tex]\(5x^2\)[/tex] to each term in [tex]\(2x + 6\)[/tex]:
[tex]\[
5x^2 \cdot 2x = 10x^3
\][/tex]
[tex]\[
5x^2 \cdot 6 = 30x^2
\][/tex]
- Distribute [tex]\(-x\)[/tex] to each term in [tex]\(2x + 6\)[/tex]:
[tex]\[
-x \cdot 2x = -2x^2
\][/tex]
[tex]\[
-x \cdot 6 = -6x
\][/tex]
- Distribute [tex]\(-3\)[/tex] to each term in [tex]\(2x + 6\)[/tex]:
[tex]\[
-3 \cdot 2x = -6x
\][/tex]
[tex]\[
-3 \cdot 6 = -18
\][/tex]
2. Combine All These Results:
Now, add all the products together:
[tex]\[
10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18
\][/tex]
3. Combine Like Terms:
Combine the like terms ([tex]\(30x^2 - 2x^2\)[/tex] and [tex]\(-6x - 6x\)[/tex]):
[tex]\[
10x^3 + (30x^2 - 2x^2) + (-6x - 6x) - 18
\][/tex]
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
So, the product of the polynomials [tex]\((5x^2 - x - 3)(2x + 6)\)[/tex] is:
[tex]\[
10x^3 + 28x^2 - 12x - 18
\][/tex]
The correct answer is:
C. [tex]\(10 x^3 + 28 x^2 - 12 x - 18\)[/tex]
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