Factoring complex trinomials (ax² + bx + c, where a ≠ 1) involves a series of steps, focusing on manipulating the trinomial to a form where you can factor by grouping. Here's a breakdown of the process:
Step-by-Step Guide to Factoring Complex Trinomials
-
Identify a, b, and c: Start by clearly identifying the coefficients a, b, and c in the trinomial ax² + bx + c.
-
*Calculate a c:* Multiply the coefficient a by the constant term c*. This product is crucial for the next steps.
-
*Find two numbers that multiply to a c and add up to b:* This is the key step. You need to find two numbers, let's call them m and n*, such that:
- m * n = a * c
- m + n = b
-
Rewrite the middle term (bx): Replace the bx term with the sum of mx and nx. The trinomial now becomes: ax² + mx + nx + c.
-
Factor by Grouping: Group the first two terms and the last two terms together: (ax² + mx) + (nx + c). Factor out the greatest common factor (GCF) from each group. Ideally, you'll have a common binomial factor remaining.
-
Factor out the Common Binomial: If you've done everything correctly, both groups will now have a common binomial factor. Factor this common binomial out of the entire expression.
Example:
Let's factor the trinomial 2x² + 7x + 3:
-
a = 2, b = 7, c = 3
-
a * c = 2 * 3 = 6
-
We need two numbers that multiply to 6 and add up to 7. These numbers are 6 and 1 (6 * 1 = 6 and 6 + 1 = 7).
-
Rewrite the middle term: 2x² + 6x + 1x + 3
-
Factor by Grouping: (2x² + 6x) + (1x + 3) = 2x(x + 3) + 1(x + 3)
-
Factor out the Common Binomial: (x + 3)(2x + 1)
Therefore, 2x² + 7x + 3 factors to (x + 3)(2x + 1).
Tips for Success:
- Practice: Factoring takes practice. The more you do it, the easier it becomes.
- Check your answer: Multiply the factors you obtain to ensure they equal the original trinomial.
- Be careful with signs: Pay close attention to the signs of a, b, and c when finding the numbers m and n.
- Don't give up! Some complex trinomials can be challenging, but persistence pays off.
