Answer :
Certainly! Let's simplify the expression [tex]\(3(x+2)(x^2-x-8)\)[/tex] step by step.
### Step 1: Distribute [tex]\((x+2)\)[/tex] to [tex]\((x^2-x-8)\)[/tex]:
To expand [tex]\((x+2)(x^2-x-8)\)[/tex], distribute each term in [tex]\((x+2)\)[/tex] to each term in [tex]\((x^2-x-8)\)[/tex]:
1. Multiply [tex]\(x\)[/tex] by each term in [tex]\((x^2-x-8)\)[/tex]:
- [tex]\(x \cdot x^2 = x^3\)[/tex]
- [tex]\(x \cdot (-x) = -x^2\)[/tex]
- [tex]\(x \cdot (-8) = -8x\)[/tex]
Result from [tex]\(x\)[/tex] is: [tex]\(x^3 - x^2 - 8x\)[/tex]
2. Multiply [tex]\(2\)[/tex] by each term in [tex]\((x^2-x-8)\)[/tex]:
- [tex]\(2 \cdot x^2 = 2x^2\)[/tex]
- [tex]\(2 \cdot (-x) = -2x\)[/tex]
- [tex]\(2 \cdot (-8) = -16\)[/tex]
Result from [tex]\(2\)[/tex] is: [tex]\(2x^2 - 2x - 16\)[/tex]
### Step 2: Combine like terms:
Combine the results from both distributions:
- [tex]\(x^3\)[/tex]
- [tex]\(-x^2 + 2x^2 = x^2\)[/tex]
- [tex]\(-8x - 2x = -10x\)[/tex]
- [tex]\(-16\)[/tex]
So, [tex]\((x+2)(x^2-x-8)\)[/tex] simplifies to:
[tex]\[ x^3 + x^2 - 10x - 16 \][/tex]
### Step 3: Distribute the 3 throughout:
Now, distribute the [tex]\(3\)[/tex] to each term in the polynomial [tex]\(x^3 + x^2 - 10x - 16\)[/tex]:
- [tex]\(3 \cdot x^3 = 3x^3\)[/tex]
- [tex]\(3 \cdot x^2 = 3x^2\)[/tex]
- [tex]\(3 \cdot (-10x) = -30x\)[/tex]
- [tex]\(3 \cdot (-16) = -48\)[/tex]
Final simplified polynomial is:
[tex]\[ 3x^3 + 3x^2 - 30x - 48 \][/tex]
Upon looking at the given options, the correct answer is:
[tex]\(3x^3 + 3x^2 - 30x - 48\)[/tex].
### Step 1: Distribute [tex]\((x+2)\)[/tex] to [tex]\((x^2-x-8)\)[/tex]:
To expand [tex]\((x+2)(x^2-x-8)\)[/tex], distribute each term in [tex]\((x+2)\)[/tex] to each term in [tex]\((x^2-x-8)\)[/tex]:
1. Multiply [tex]\(x\)[/tex] by each term in [tex]\((x^2-x-8)\)[/tex]:
- [tex]\(x \cdot x^2 = x^3\)[/tex]
- [tex]\(x \cdot (-x) = -x^2\)[/tex]
- [tex]\(x \cdot (-8) = -8x\)[/tex]
Result from [tex]\(x\)[/tex] is: [tex]\(x^3 - x^2 - 8x\)[/tex]
2. Multiply [tex]\(2\)[/tex] by each term in [tex]\((x^2-x-8)\)[/tex]:
- [tex]\(2 \cdot x^2 = 2x^2\)[/tex]
- [tex]\(2 \cdot (-x) = -2x\)[/tex]
- [tex]\(2 \cdot (-8) = -16\)[/tex]
Result from [tex]\(2\)[/tex] is: [tex]\(2x^2 - 2x - 16\)[/tex]
### Step 2: Combine like terms:
Combine the results from both distributions:
- [tex]\(x^3\)[/tex]
- [tex]\(-x^2 + 2x^2 = x^2\)[/tex]
- [tex]\(-8x - 2x = -10x\)[/tex]
- [tex]\(-16\)[/tex]
So, [tex]\((x+2)(x^2-x-8)\)[/tex] simplifies to:
[tex]\[ x^3 + x^2 - 10x - 16 \][/tex]
### Step 3: Distribute the 3 throughout:
Now, distribute the [tex]\(3\)[/tex] to each term in the polynomial [tex]\(x^3 + x^2 - 10x - 16\)[/tex]:
- [tex]\(3 \cdot x^3 = 3x^3\)[/tex]
- [tex]\(3 \cdot x^2 = 3x^2\)[/tex]
- [tex]\(3 \cdot (-10x) = -30x\)[/tex]
- [tex]\(3 \cdot (-16) = -48\)[/tex]
Final simplified polynomial is:
[tex]\[ 3x^3 + 3x^2 - 30x - 48 \][/tex]
Upon looking at the given options, the correct answer is:
[tex]\(3x^3 + 3x^2 - 30x - 48\)[/tex].
