Answer :
To simplify the expression [tex]\(3(x+2)(x^2-x-8)\)[/tex], we need to follow these steps:
1. Distribute the 3: Start by distributing the 3 across the first part of the expression. This means we multiply each term inside the parenthesis [tex]\((x+2)\)[/tex] by 3:
[tex]\[
3(x+2) = 3x + 6
\][/tex]
2. Expand the expression: Next, distribute each term in [tex]\((3x + 6)\)[/tex] across each term in the polynomial [tex]\((x^2 - x - 8)\)[/tex]:
- Multiply [tex]\(3x\)[/tex] by each term in [tex]\((x^2 - x - 8)\)[/tex]:
[tex]\[
(3x) \cdot (x^2 - x - 8) = 3x \cdot x^2 + 3x \cdot (-x) + 3x \cdot (-8)
\][/tex]
[tex]\[
= 3x^3 - 3x^2 - 24x
\][/tex]
- Multiply [tex]\(6\)[/tex] by each term in [tex]\((x^2 - x - 8)\)[/tex]:
[tex]\[
6 \cdot (x^2 - x - 8) = 6 \cdot x^2 + 6 \cdot (-x) + 6 \cdot (-8)
\][/tex]
[tex]\[
= 6x^2 - 6x - 48
\][/tex]
3. Combine like terms: Now, add together the results from the distributions:
[tex]\[
3x^3 - 3x^2 - 24x + 6x^2 - 6x - 48
\][/tex]
Combine the like terms:
- For [tex]\(x^3\)[/tex]: [tex]\(3x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(-3x^2 + 6x^2 = 3x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-24x - 6x = -30x\)[/tex]
- Constant term: [tex]\(-48\)[/tex]
So, the simplified expression is:
[tex]\[
3x^3 + 3x^2 - 30x - 48
\][/tex]
Therefore, the expression [tex]\(3(x+2)(x^2-x-8)\)[/tex] simplifies to [tex]\(3x^3 + 3x^2 - 30x - 48\)[/tex].
1. Distribute the 3: Start by distributing the 3 across the first part of the expression. This means we multiply each term inside the parenthesis [tex]\((x+2)\)[/tex] by 3:
[tex]\[
3(x+2) = 3x + 6
\][/tex]
2. Expand the expression: Next, distribute each term in [tex]\((3x + 6)\)[/tex] across each term in the polynomial [tex]\((x^2 - x - 8)\)[/tex]:
- Multiply [tex]\(3x\)[/tex] by each term in [tex]\((x^2 - x - 8)\)[/tex]:
[tex]\[
(3x) \cdot (x^2 - x - 8) = 3x \cdot x^2 + 3x \cdot (-x) + 3x \cdot (-8)
\][/tex]
[tex]\[
= 3x^3 - 3x^2 - 24x
\][/tex]
- Multiply [tex]\(6\)[/tex] by each term in [tex]\((x^2 - x - 8)\)[/tex]:
[tex]\[
6 \cdot (x^2 - x - 8) = 6 \cdot x^2 + 6 \cdot (-x) + 6 \cdot (-8)
\][/tex]
[tex]\[
= 6x^2 - 6x - 48
\][/tex]
3. Combine like terms: Now, add together the results from the distributions:
[tex]\[
3x^3 - 3x^2 - 24x + 6x^2 - 6x - 48
\][/tex]
Combine the like terms:
- For [tex]\(x^3\)[/tex]: [tex]\(3x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(-3x^2 + 6x^2 = 3x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-24x - 6x = -30x\)[/tex]
- Constant term: [tex]\(-48\)[/tex]
So, the simplified expression is:
[tex]\[
3x^3 + 3x^2 - 30x - 48
\][/tex]
Therefore, the expression [tex]\(3(x+2)(x^2-x-8)\)[/tex] simplifies to [tex]\(3x^3 + 3x^2 - 30x - 48\)[/tex].
