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Answer :
To simplify the expression [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex], follow these steps:
1. Distribute each term in the first polynomial across the second polynomial:
We'll use the distributive property to expand the expression:
[tex]\[
(6x - 5)(2x^2 - 3x - 6) = 6x \cdot (2x^2 - 3x - 6) - 5 \cdot (2x^2 - 3x - 6)
\][/tex]
2. Multiply each part:
- For [tex]\(6x \cdot (2x^2 - 3x - 6)\)[/tex]:
- [tex]\(6x \cdot 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \cdot -3x = -18x^2\)[/tex]
- [tex]\(6x \cdot -6 = -36x\)[/tex]
The result of this part is: [tex]\(12x^3 - 18x^2 - 36x\)[/tex]
- For [tex]\(-5 \cdot (2x^2 - 3x - 6)\)[/tex]:
- [tex]\(-5 \cdot 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \cdot -3x = 15x\)[/tex]
- [tex]\(-5 \cdot -6 = 30\)[/tex]
The result of this part is: [tex]\(-10x^2 + 15x + 30\)[/tex]
3. Combine the results:
Add the simplified parts together:
[tex]\[
12x^3 - 18x^2 - 36x - 10x^2 + 15x + 30
\][/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\(12x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-36x + 15x = -21x\)[/tex]
- The constant term: [tex]\(+30\)[/tex]
This gives us the simplified expression:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
Therefore, the correct simplification of [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex] is [tex]\(12x^3 - 28x^2 - 21x + 30\)[/tex].
1. Distribute each term in the first polynomial across the second polynomial:
We'll use the distributive property to expand the expression:
[tex]\[
(6x - 5)(2x^2 - 3x - 6) = 6x \cdot (2x^2 - 3x - 6) - 5 \cdot (2x^2 - 3x - 6)
\][/tex]
2. Multiply each part:
- For [tex]\(6x \cdot (2x^2 - 3x - 6)\)[/tex]:
- [tex]\(6x \cdot 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \cdot -3x = -18x^2\)[/tex]
- [tex]\(6x \cdot -6 = -36x\)[/tex]
The result of this part is: [tex]\(12x^3 - 18x^2 - 36x\)[/tex]
- For [tex]\(-5 \cdot (2x^2 - 3x - 6)\)[/tex]:
- [tex]\(-5 \cdot 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \cdot -3x = 15x\)[/tex]
- [tex]\(-5 \cdot -6 = 30\)[/tex]
The result of this part is: [tex]\(-10x^2 + 15x + 30\)[/tex]
3. Combine the results:
Add the simplified parts together:
[tex]\[
12x^3 - 18x^2 - 36x - 10x^2 + 15x + 30
\][/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\(12x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-36x + 15x = -21x\)[/tex]
- The constant term: [tex]\(+30\)[/tex]
This gives us the simplified expression:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
Therefore, the correct simplification of [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex] is [tex]\(12x^3 - 28x^2 - 21x + 30\)[/tex].
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