Answer :
To simplify the expression [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex], we can use the distributive property, also known as the FOIL method for binomials, to expand the terms. Here's how you can do it step-by-step:
1. Distribute the First Term:
- Multiply [tex]\(6x\)[/tex] by each of the terms in the second expression.
- [tex]\(6x \cdot 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \cdot -3x = -18x^2\)[/tex]
- [tex]\(6x \cdot -6 = -36x\)[/tex]
2. Distribute the Second Term:
- Multiply [tex]\(-5\)[/tex] by each of the terms in the second expression.
- [tex]\(-5 \cdot 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \cdot -3x = 15x\)[/tex]
- [tex]\(-5 \cdot -6 = 30\)[/tex]
3. Combine All the Terms:
- After expanding, write out all the terms:
[tex]\[
12x^3 - 18x^2 - 36x - 10x^2 + 15x + 30
\][/tex]
4. Combine Like Terms:
- Combine terms with the same degree (the same power of [tex]\(x\)[/tex]):
- For [tex]\(x^3\)[/tex]: [tex]\(12x^3\)[/tex] (there’s only one)
- For [tex]\(x^2\)[/tex]: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-36x + 15x = -21x\)[/tex]
- Constant term: [tex]\(30\)[/tex]
5. Write the Final Expression:
- The simplified expression is:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
Therefore, the correct simplification of the expression [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex] is:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
This corresponds to the choice:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
1. Distribute the First Term:
- Multiply [tex]\(6x\)[/tex] by each of the terms in the second expression.
- [tex]\(6x \cdot 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \cdot -3x = -18x^2\)[/tex]
- [tex]\(6x \cdot -6 = -36x\)[/tex]
2. Distribute the Second Term:
- Multiply [tex]\(-5\)[/tex] by each of the terms in the second expression.
- [tex]\(-5 \cdot 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \cdot -3x = 15x\)[/tex]
- [tex]\(-5 \cdot -6 = 30\)[/tex]
3. Combine All the Terms:
- After expanding, write out all the terms:
[tex]\[
12x^3 - 18x^2 - 36x - 10x^2 + 15x + 30
\][/tex]
4. Combine Like Terms:
- Combine terms with the same degree (the same power of [tex]\(x\)[/tex]):
- For [tex]\(x^3\)[/tex]: [tex]\(12x^3\)[/tex] (there’s only one)
- For [tex]\(x^2\)[/tex]: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-36x + 15x = -21x\)[/tex]
- Constant term: [tex]\(30\)[/tex]
5. Write the Final Expression:
- The simplified expression is:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
Therefore, the correct simplification of the expression [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex] is:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
This corresponds to the choice:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
