Factor polynomials with four terms using grouping

The first thing we will always do when factoring is try to factor out a GCF. This GCF is often a monomial like in the problem 5xy + 10xz the GCF is the monomial 5x, so we would have 5x(y + 2z). However, a GCF does not have to be a monomial, it could be a binomial. To see this, consider the following two example.

Example 1:

Factoring 3ax -7bx

Solution:

Both have x in common, factor it out

Our solution is x(3a -7b).

Now the same problem, but instead of x we have (2a +5b).

Example 2:

Factoring 3a(2a +5b) -7b(2a +5b)

Solution:

Both have (2a +5b) in common, factor it out

Our solution is

(2a +5b)(3a -7b)

In the same way we factored out a GCF of x we can factor out a GCF which is a binomial, (2a + 5b). This process can be extended to factor problems where there is no GCF to factor out, or after the GCF is factored out, there is more factoring that can be done. Here we will have to use another strategy to factor. We will use a process known as grouping. Grouping is how we will factor if there are four terms in the problem. Remember, factoring is like multiplying in reverse, so first we will look at a multiplication problem and then try to reverse the process.

Also Read : Inequalities of linear equation

Example 3:

Solve (2a +3)(5b +2)

Solution:

Distribute (2a +3) into second parenthesis

5b(2a +3) +2(2a +3)

Distribute each monomial

Our solution is

10ab +15b +4a +6

The solution has four terms in it. We arrived at the solution by looking at the two parts, 5b(2a + 3) and 2(2a + 3). When we are factoring by grouping we will always divide the problem into two parts, the first two terms and the last two terms. Then we can factor the GCF out of both the left and right sides. When we do this our hope is what is left in the parenthesis will match on both the left and right. If they match we can pull this matching GCF out front, putting the rest in parenthesis and we will be factored. The next example is the same problem

worked backwards, factoring instead of multiplying.

Example 4:

Factorize 10ab +15b +4a +6

Solution:

Split problem into two groups

GCF on left is 5b, on the right is 2

(2a +3) is the same! Factor out this GCF

Our solution is (2a +3)(5b +2)

The key for grouping to work is after the GCF is factored out of the left and right, the two binomials must match exactly. If there is any difference between the two we either have to do some adjusting or it can’t be factored using the grouping method. Consider the following example.

Example 5:

Factorize 6x + 9xy – 14x – 21y.

Solution:

Split problem into two groups

GCF on left is 3x, on right is 7

The signs in the parenthesis don’t match!

when the signs don’t match on both terms we can easily make them match by factoring the opposite of the GCF on the right side. Instead of 7 we will use – 7. This will change the signs inside the second parenthesis.

(2x +3y) is the same! Factor out this GCF

Our solution is (2x + 3y)(3x – 7).

Often we can recognize early that we need to use the opposite of the GCF when factoring. If the first term of the first binomial is positive in the problem, we will also want the first term of the second binomial to be positive. If it is negative then we will use the opposite of the GCF to be sure they match.

Example 6:

Factorize 5xy – 8x – 10y +16

Solution:

Split the problem into two groups

GCF on left is x, on right we need a negative, so we use -2

(5y – 8) is the same! Factor out this GCF

Our solution is (5y – 8)(x – 2).

Sometimes when factoring the GCF out of the left or right side there is no GCF to factor out. In this case we will use either the GCF of 1 or -1. Often this is all we need to be sure the two binomials match.

Example 7:

Factorize 12ab -14a -6b +7

Solution:

Split the problem into two groups

GCF on left is 2a, on right, no GCF, use -1

(6b -7) is the same! Factor out this GCF

Our solution is (6b – 7)(2a – 1).

Example 8:

Factorize : $6x^3-15x^2+2x-5$

Solution:

Split problem into two groups

GCF on left is $3x^2$, on right, no GCF, use 1

(2x -5) is the same! Factor out this GCF

Our solution is $(2x-5)(3x^2+1)$.

World View Note: Sofia Kovalevskaya of Russia was the ﬁrst woman on the editorial staff of a mathematical journal in the late 19th century. She also did research on how the rings of Saturn rotated.

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