How To Factor Trinomials A Not 1

How to Factor Trinomials When a ≠ 1: A Step-by-Step Guide

Factoring trinomials is a crucial skill in algebra. While factoring when the leading coefficient (the 'a' in ax² + bx + c) is 1 is relatively straightforward, factoring when 'a' is not equal to 1 presents a greater challenge. This comprehensive guide will break down the process, equipping you with the knowledge and confidence to tackle these more complex trinomials.

Understanding the Problem: ax² + bx + c where a ≠ 1

When factoring trinomials of the form ax² + bx + c, where 'a' is not 1, we're looking for two binomials whose product equals the original trinomial. This is more involved than when a=1 because we need to consider factors of both 'a' and 'c'. Simply finding factors that add up to 'b' is insufficient.

The Method: Factoring by Grouping

The most common and reliable method for factoring trinomials where a ≠ 1 is factoring by grouping. Here's a step-by-step approach:

Step 1: Find the Product ac

First, multiply the coefficient of the x² term ('a') by the constant term ('c'). This gives you the product 'ac'.

Example: Let's factor 2x² + 7x + 3. Here, a = 2 and c = 3. Therefore, ac = 2 * 3 = 6.

Step 2: Find Two Numbers That Add Up to 'b' and Multiply to 'ac'

Next, find two numbers that add up to the coefficient of the x term ('b') and multiply to the product 'ac' you calculated in Step 1.

Example (continued): We need two numbers that add up to 7 (our 'b') and multiply to 6 (our 'ac'). These numbers are 6 and 1 (6 + 1 = 7 and 6 * 1 = 6).

Step 3: Rewrite the Trinomial

Rewrite the original trinomial, replacing the 'bx' term with the two numbers you found in Step 2. Express these numbers as coefficients of x.

Example (continued): Our original trinomial was 2x² + 7x + 3. We rewrite it as 2x² + 6x + 1x + 3.

Step 4: Factor by Grouping

Now, group the first two terms and the last two terms together. Factor out the greatest common factor (GCF) from each group.

Example (continued):

• (2x² + 6x) + (1x + 3)
• 2x(x + 3) + 1(x + 3)

Step 5: Factor Out the Common Binomial

Notice that both terms now have a common binomial factor: (x + 3). Factor this out.

Example (continued):

• (x + 3)(2x + 1)

This is your factored form!

Checking Your Answer

Always check your answer by expanding the factored form using the FOIL method (First, Outer, Inner, Last). If you get back to your original trinomial, you've factored correctly.

Example (continued): (x + 3)(2x + 1) = 2x² + x + 6x + 3 = 2x² + 7x + 3. This matches our original trinomial, so our factoring is correct.

Practicing Makes Perfect

Factoring trinomials when a ≠ 1 takes practice. Start with simpler examples and gradually work your way up to more challenging ones. The more you practice, the more comfortable and proficient you'll become. Remember to always check your work! Mastering this skill will significantly improve your ability to solve more complex algebraic problems.