To factor the polynomial [tex]\( x^3 + 8x^2 + 15x \)[/tex] completely, follow these detailed steps:
### Step 1: Identify and Factor Out the Greatest Common Factor (GCF)
First, observe that each term in the polynomial [tex]\( x^3 + 8x^2 + 15x \)[/tex] contains the variable [tex]\( x \)[/tex]. Therefore, the greatest common factor (GCF) of these terms is [tex]\( x \)[/tex].
Factor out the GCF ([tex]\( x \)[/tex]) from each term:
[tex]\[ x^3 + 8x^2 + 15x = x(x^2 + 8x + 15) \][/tex]
### Step 2: Factor the Quadratic Expression Inside the Parenthesis
Next, focus on factoring the quadratic expression inside the parenthesis: [tex]\( x^2 + 8x + 15 \)[/tex].
To factor [tex]\( x^2 + 8x + 15 \)[/tex], we need to find two numbers that:
1. Multiply to the constant term, 15.
2. Add up to the coefficient of the middle term, 8.
The pairs of factors of 15 are:
- [tex]\( 1 \times 15 \)[/tex]
- [tex]\( 3 \times 5 \)[/tex]
Among these pairs, the numbers 3 and 5 add up to 8.
So, we can write the quadratic expression as the product of two binomials:
[tex]\[ x^2 + 8x + 15 = (x + 3)(x + 5) \][/tex]
### Step 3: Combine the Factorizations
Now, combine the GCF with the factored form of the quadratic expression:
[tex]\[ x(x^2 + 8x + 15) = x(x + 3)(x + 5) \][/tex]
### Conclusion
The completely factored form of the polynomial [tex]\( x^3 + 8x^2 + 15x \)[/tex] is:
[tex]\[ \boxed{x(x + 3)(x + 5)} \][/tex]
