Answer :
Sure! Let's simplify the expression [tex]\(3(x+2)(x^2-x-8)\)[/tex] step-by-step:
1. Distribute within the parentheses:
To simplify the expression, we first need to expand [tex]\((x+2)(x^2-x-8)\)[/tex].
- Distribute [tex]\(x\)[/tex] over [tex]\((x^2 - x - 8)\)[/tex]:
[tex]\[
x \cdot (x^2) = x^3, \quad x \cdot (-x) = -x^2, \quad x \cdot (-8) = -8x
\][/tex]
So, multiplying [tex]\(x\)[/tex] gives: [tex]\(x^3 - x^2 - 8x\)[/tex].
- Distribute [tex]\(2\)[/tex] over [tex]\((x^2 - x - 8)\)[/tex]:
[tex]\[
2 \cdot (x^2) = 2x^2, \quad 2 \cdot (-x) = -2x, \quad 2 \cdot (-8) = -16
\][/tex]
So, multiplying 2 gives: [tex]\(2x^2 - 2x - 16\)[/tex].
2. Combine like terms:
Now combine all the terms from both distributions:
[tex]\[
x^3 - x^2 - 8x + 2x^2 - 2x - 16
\][/tex]
Combine the like terms:
[tex]\[
x^3 + (-x^2 + 2x^2) - (8x + 2x) - 16
\][/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-x^2 + 2x^2 = x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-8x - 2x = -10x\)[/tex]
So, the combined result is:
[tex]\[
x^3 + x^2 - 10x - 16
\][/tex]
3. Multiply by outer 3:
Now, we multiply the entire expression [tex]\((x^3 + x^2 - 10x - 16)\)[/tex] by [tex]\(3\)[/tex].
Distribute the [tex]\(3\)[/tex]:
[tex]\[
3 \cdot x^3 = 3x^3, \quad 3 \cdot x^2 = 3x^2, \quad 3 \cdot (-10x) = -30x, \quad 3 \cdot (-16) = -48
\][/tex]
Combining these gives:
[tex]\[
3x^3 + 3x^2 - 30x - 48
\][/tex]
So, the simplified expression is [tex]\(\boxed{3x^3 + 3x^2 - 30x - 48}\)[/tex].
1. Distribute within the parentheses:
To simplify the expression, we first need to expand [tex]\((x+2)(x^2-x-8)\)[/tex].
- Distribute [tex]\(x\)[/tex] over [tex]\((x^2 - x - 8)\)[/tex]:
[tex]\[
x \cdot (x^2) = x^3, \quad x \cdot (-x) = -x^2, \quad x \cdot (-8) = -8x
\][/tex]
So, multiplying [tex]\(x\)[/tex] gives: [tex]\(x^3 - x^2 - 8x\)[/tex].
- Distribute [tex]\(2\)[/tex] over [tex]\((x^2 - x - 8)\)[/tex]:
[tex]\[
2 \cdot (x^2) = 2x^2, \quad 2 \cdot (-x) = -2x, \quad 2 \cdot (-8) = -16
\][/tex]
So, multiplying 2 gives: [tex]\(2x^2 - 2x - 16\)[/tex].
2. Combine like terms:
Now combine all the terms from both distributions:
[tex]\[
x^3 - x^2 - 8x + 2x^2 - 2x - 16
\][/tex]
Combine the like terms:
[tex]\[
x^3 + (-x^2 + 2x^2) - (8x + 2x) - 16
\][/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-x^2 + 2x^2 = x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-8x - 2x = -10x\)[/tex]
So, the combined result is:
[tex]\[
x^3 + x^2 - 10x - 16
\][/tex]
3. Multiply by outer 3:
Now, we multiply the entire expression [tex]\((x^3 + x^2 - 10x - 16)\)[/tex] by [tex]\(3\)[/tex].
Distribute the [tex]\(3\)[/tex]:
[tex]\[
3 \cdot x^3 = 3x^3, \quad 3 \cdot x^2 = 3x^2, \quad 3 \cdot (-10x) = -30x, \quad 3 \cdot (-16) = -48
\][/tex]
Combining these gives:
[tex]\[
3x^3 + 3x^2 - 30x - 48
\][/tex]
So, the simplified expression is [tex]\(\boxed{3x^3 + 3x^2 - 30x - 48}\)[/tex].
