Understanding Factoring
Factoring is the process of breaking down an expression into simpler components, often called factors. This allows us to simplify expressions or solve equations more easily.
Given Expression
The expression to factor is: 2x^3 + 5x^2y - 12xy^2.
Step 1: Identify Common Factors
- Look for common factors in each term.
- The terms are:
- 2x^3
- 5x^2y
- -12xy^2
- The common factor across all terms is "xy".
Step 2: Factor Out the Common Factor
- Factor "xy" from each term:
- 2x^3 = 2x^2 * xy
- 5x^2y = 5x * xy
- -12xy^2 = -12 * xy
- Thus, we can rewrite the expression as:
- xy(2x^2 + 5x - 12)
Step 3: Factoring the Quadratic
Now, we need to factor the quadratic: 2x^2 + 5x - 12.
- Look for two numbers that multiply to (2 * -12 = -24) and add to 5.
- The numbers are 8 and -3.
Step 4: Rewrite and Factor Further
- Rewrite the quadratic:
- 2x^2 + 8x - 3x - 12
- Group the terms:
- (2x^2 + 8x) + (-3x - 12)
- Factor by grouping:
- 2x(x + 4) - 3(x + 4)
- Now we can factor out (x + 4):
- (x + 4)(2x - 3)
Final Factored Form
Combining the common factor gives us the complete factored form:
- xy(x + 4)(2x - 3)
This is the fully factored expression. Understanding these steps will greatly enhance your algebra skills!