(x%5E3%2B5x%5E2-6x).jpeg "(x^2+8x+12)(x^3+5x^2-6x)")
Factoring and Simplifying the Expression: (x^2+8x+12)(x^3+5x^2-6x)
This expression involves multiplying two polynomials. To simplify it, we can factor both polynomials and then multiply the resulting expressions.
Factoring the Polynomials:
1. Factoring (x^2 + 8x + 12)
- Find two numbers that add up to 8 and multiply to 12. These numbers are 6 and 2.
- Therefore, we can factor this expression as: (x + 6)(x + 2)
2. Factoring (x^3 + 5x^2 - 6x)
- First, factor out the greatest common factor (GCF) which is x: x(x^2 + 5x - 6)
- Now, factor the quadratic expression inside the parentheses. Find two numbers that add up to 5 and multiply to -6. These numbers are 6 and -1.
- Therefore, we can factor this expression as: x(x + 6)(x - 1)
Multiplying the Factored Expressions:
Now that we have factored both polynomials, we can multiply them together:
(x + 6)(x + 2) * x(x + 6)(x - 1)
We can rearrange the factors for easier multiplication:
x(x + 6)(x + 6)(x + 2)(x - 1)
This simplifies to:
x(x + 6)^2 (x + 2)(x - 1)
Final Simplified Expression:
Therefore, the simplified expression for (x^2 + 8x + 12)(x^3 + 5x^2 - 6x) is: x(x + 6)^2 (x + 2)(x - 1)