Sure, let's solve the problem step-by-step by factoring the given expression [tex]\(15x^3 + 3x^2 - 3x\)[/tex].
### Step 1: Identify the Common Factor in Each Term
First, we look for the greatest common factor (GCF) among the terms of the expression.
1. In the term [tex]\(15x^3\)[/tex]:
- Coefficient: 15
- Variable part: [tex]\(x^3\)[/tex]
2. In the term [tex]\(3x^2\)[/tex]:
- Coefficient: 3
- Variable part: [tex]\(x^2\)[/tex]
3. In the term [tex]\(-3x\)[/tex]:
- Coefficient: -3
- Variable part: [tex]\(x\)[/tex]
The GCF of the coefficients (15, 3, -3) is 3. For the variable part, the lowest power of [tex]\(x\)[/tex] across the terms is [tex]\(x\)[/tex].
So, the GCF of the entire expression is [tex]\(3x\)[/tex].
### Step 2: Factor Out the Common Factor
We now divide each term by [tex]\(3x\)[/tex] and rewrite the expression:
1. [tex]\(15x^3 \div 3x = 5x^2\)[/tex]
2. [tex]\(3x^2 \div 3x = x\)[/tex]
3. [tex]\(-3x \div 3x = -1\)[/tex]
This gives us the expression:
[tex]\[15x^3 + 3x^2 - 3x = 3x(5x^2 + x - 1)\][/tex]
### Step 3: Verify the Factored Expression
To make sure we've factored correctly, we can distribute [tex]\(3x\)[/tex] back through the expression inside the parentheses:
[tex]\[3x(5x^2) + 3x(x) - 3x(1) = 15x^3 + 3x^2 - 3x\][/tex]
Since this matches the original expression, our factorization is correct.
Thus, the factored form of the expression [tex]\(15x^3 + 3x^2 - 3x\)[/tex] is:
[tex]\[3x(5x^2 + x - 1)\][/tex]
This is the final answer.
