Answer :
To simplify the expression [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex], follow these steps:
1. Distribute Each Term:
First, distribute [tex]\(6x\)[/tex] to each term in the second polynomial [tex]\(2x^2 - 3x - 6\)[/tex]:
- [tex]\(6x \cdot 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \cdot (-3x) = -18x^2\)[/tex]
- [tex]\(6x \cdot (-6) = -36x\)[/tex]
So, [tex]\(6x(2x^2 - 3x - 6) = 12x^3 - 18x^2 - 36x\)[/tex].
2. Distribute [tex]\(-5\)[/tex] to Each Term:
Next, distribute [tex]\(-5\)[/tex] to each term in the second polynomial:
- [tex]\(-5 \cdot 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \cdot (-3x) = 15x\)[/tex]
- [tex]\(-5 \cdot (-6) = 30\)[/tex]
So, [tex]\(-5(2x^2 - 3x - 6) = -10x^2 + 15x + 30\)[/tex].
3. Combine the Results:
Now, add together the results from step 1 and step 2:
[tex]\[
(12x^3 - 18x^2 - 36x) + (-10x^2 + 15x + 30)
\][/tex]
- 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].
Thus, the simplified expression is:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
Therefore, the correct simplification is [tex]\(\boxed{12x^3 - 28x^2 - 21x + 30}\)[/tex].
1. Distribute Each Term:
First, distribute [tex]\(6x\)[/tex] to each term in the second polynomial [tex]\(2x^2 - 3x - 6\)[/tex]:
- [tex]\(6x \cdot 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \cdot (-3x) = -18x^2\)[/tex]
- [tex]\(6x \cdot (-6) = -36x\)[/tex]
So, [tex]\(6x(2x^2 - 3x - 6) = 12x^3 - 18x^2 - 36x\)[/tex].
2. Distribute [tex]\(-5\)[/tex] to Each Term:
Next, distribute [tex]\(-5\)[/tex] to each term in the second polynomial:
- [tex]\(-5 \cdot 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \cdot (-3x) = 15x\)[/tex]
- [tex]\(-5 \cdot (-6) = 30\)[/tex]
So, [tex]\(-5(2x^2 - 3x - 6) = -10x^2 + 15x + 30\)[/tex].
3. Combine the Results:
Now, add together the results from step 1 and step 2:
[tex]\[
(12x^3 - 18x^2 - 36x) + (-10x^2 + 15x + 30)
\][/tex]
- 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].
Thus, the simplified expression is:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
Therefore, the correct simplification is [tex]\(\boxed{12x^3 - 28x^2 - 21x + 30}\)[/tex].
