2Factor by GroupingWhen polynomials contain four terms, it is sometimes easier to group like terms in order to factor.Your goal is to create a common factor.You can also move terms around in the polynomial to create a common factor.Practice makes you better in recognizing common factors.
4Factor by Grouping Example 1: FACTOR: 3xy - 21y + 5x – 35Factor the first two terms:3xy - 21y = 3y (x – 7)Factor the last two terms:+ 5x - 35 = 5 (x – 7)The green parentheses are the same so it’s the common factorNow you have a common factor(x - 7) (3y + 5)
5Factor by Grouping Example 2: FACTOR: 6mx – 4m + 3rx – 2rFactor the first two terms:6mx – 4m = 2m (3x - 2)Factor the last two terms:+ 3rx – 2r = r (3x - 2)The green parentheses are the same so it’s the common factorNow you have a common factor(3x - 2) (2m + r)
6Factor by Grouping Example 3: FACTOR: 15x – 3xy + 4y –20Factor the first two terms:15x – 3xy = 3x (5 – y)Factor the last two terms:+ 4y –20 = 4 (y – 5)The green parentheses are opposites so change the sign on the 4- 4 (-y + 5) or – 4 (5 - y)Now you have a common factor(5 – y) (3x – 4)
8Factoring Trinominals When trinomials have a degree of “2”, they are known as quadratics.We learned earlier to use the “diamond” to factor trinomials that had a “1” in front of the squared term.x2 + 12x + 35(x + 7)(x + 5)
9More Factoring Trinomials When there is a coefficient larger than “1” in front of the squared term, we can use a modified diamond or square to find the factors.Always remember to look for a GCF before you do ANY other factoring.
10More Factoring Trinomials Let’s try this example3x2 + 13x + 4Make a boxWrite the factors of the first term.Write the factors of the last term.Multiply on the diagonal and add to see if you get the middle term of the trinomial. If so, you’re done!
12Difference of SquaresWhen factoring using a difference of squares, look for the following three things:only 2 termsminus sign between themboth terms must be perfect squaresIf all 3 of the above are true, write two( ), one with a + sign and one with a – sign : ( ) ( ).
15Perfect Square Trinomials When factoring using perfect square trinomials, look for the following three things:3 termslast term must be positivefirst and last terms must be perfect squaresIf all three of the above are true, write one ( )2 using the sign of the middle term.
18Factoring CompletelyNow that we’ve learned all the types of factoring, we need to remember to use them all.Whenever it says to factor, you must break down the expression into the smallest possiblefactors.Let’s review all the ways to factor.
19Types of Factoring Look for GCF first. Count the number of terms: 4 terms – factor by grouping3 terms -look for perfect square trinomialif not, try diamond or box2 terms -look for difference of squaresIf any ( ) still has an exponent of 2 or more, see if you can factor again.
21Steps to Solve Equations by Factoring We know that an equation must be solved for the unknown.Up to now, we have only solved equations with a degree of 1.2x + 8 = 4x +6-2x + 8 = 6-2x = -2x = 1
22Steps to Solve Equations by Factoring If an equation has a degree of 2 or higher, we cannot solve it until it has been factored.You must first get “0” on one side of the = sign before you try any factoring.Once you have “0” on one side, use all your rules for factoring to make 2 ( ) or factors.
23Steps to Solve Equations by Factoring Next, set each factor = 0 and solve for the unknown.x2 + 12x = Factor GCFx(x + 12) = 0(set each factor = 0, & solve)x = x + 12 = 0x = - 12You now have 2 answers, x = 0 and x = -12.