How Do You Factor Polynomials by Grouping?

# How Do You Factor Polynomials by Grouping?

Factoring ax²+bx+c by grouping has some similarity to factoring into two binomials, but grouping works better when a and c are both not equal to 1. The factoring becomes more intricate when there are two constants to factor separately.

1. Determine ac and a+c

Both the terms ac and a+c are required. As a simple example, 6x²+7x+1 shows ac = 6 and a+c = 7.

2. Factor ac into possible multiples

In this case, 6 can factor into 6x1 or 3x2.

3. Find a factor pair of ac that equals b

If ac is positive, both terms are positive or negative based on the sign of b. If it is negative, one term is positive while the other is negative in order to equal b. In the example, 6x1 = 7, so this is the factor pair that is desired.

4. Split b into the factor pairs in the bx term of the polynomial

So, 6x²+7x+1 = 6x²+6x+x+1. Sometimes, the middle terms must be reversed to find the best match for the next step.

5. Factor as much as possible out of each binomial and factor by group.

Now 6x²+6x+x+1 = 6x(x+1)+1(x+1). The final step is factoring the binomial (x+1) from each term, giving (6x+1)(x+1) as the final answer.

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