Investigation: Factor by Grouping

Investigation: Tougher Factoring

Part 1: Using the guess and check method.

Step 1: Simplify (2x + 3)(3x – 5)

So far, we have only factored problems that were either special, or the leading coefficient has been 1 (x2 + bx + c). Today, our leading coefficient will be greater than one (ax2 + bx + c)

Notice how there is not a direct relationship between b and c in the above problem.

Step 2: Guess and check the factoring. This way just takes persistence, so keep on trying until it works! Use pencil, so you don’t have to cross out over and over!

Example: Factor 2x2 + 11x + 12

·  You know that the expression will factor into (___x __ ___)(___x ______)

·  What kind of signs must go in the parentheses?

·  Now look at the “a.” What are the factors of it? Those must go in the front of our parentheses.

·  Now look at the “c”. What are the factors of it? Here is where the guess and check comes in. Try a pair of factors. Then check to see if the “inners” and “outers” will add together to get b.

·  Continue to guess and check until you have it factored.

2x2 + 11x + 12 = ( )( )

Example: Factor 3x2 +13x – 10

Example: Factor 2x2 - 7x + 6

Example: Factor 2x2 + 7x + 6

Example: Factor 4x2 + 7x + 3

Example: Factor 6x2 + 5x – 6

Part 2: Use the Grouping – Triple G “Group, GCF, GCF”

Guess and check is useful when there are not many factors of a and c.

But when a or c have lots of factors, and lots of pairs to try, sometimes it may be useful to know another method.

Step 1: Factor out the GCF:

a. 4(x + 2) – 5x(x + 2)

b. 2a(b – 3) + 5(b – 3)

Now let’s try to factor x3 – 5x2 – x + 5 by grouping.

Step 1: Rearrange the terms so that you have two pairs of terms that each have a GCF. Put parentheses around them.

Step 2: Factor out the GCF of each. That should leave you with the same thing in parentheses for each pair.

Step 3: Factor out the binomial GCF.

Try These:

a. x6 + 2x4 -16x2 -32 b. x9 – x6 – x3 + 1 c. 4x3- x2 -4x + 1

d. 10x3 – 8x2 + 25x – 20 e. 27x9 + x6 -27x3 -1

Now let’s try some trinomials by grouping.

Remember: These trinomials came from 4 terms where we just combined the like terms to get that middle term.

Step 4: First simplify: (3x + 5)(2x – 3)

Notice how the middle two terms combined to get the middle term in the final answer.

Step 5: So let’s try to factor 2x2 + 13x + 20 by grouping.

·  Multiply the a and the c term.

·  Find 2 numbers that multiply to that number and add to the b. Watch the positives and negatives!

·  So instead of writing 13x, we can break it down into ___x + ___x

2x2 + 13x + 20

= 2x2 + ____x + ____x + 20

·  Now group together the pairs like we did before and factor!

Step 6: Let’s try another together.

Factor 2x2 – 19x + 24

Step 7: Try these by grouping:

a. 4x8 - 61x4 + 225 b. 5x2 + 24x – 5 c. 5x2 + 29x + 20

d. 4x2 + 4x – 15 e. 4x2 - 4x – 15 f. 8x4 + 10x2 – 3

g. 2x4 + x2 – 6 h. x2 – 7x – 18