Factoring Polynomials Grouping, Trinomials, Binomials, GCF ,Quadratic form & Solving Equations

Student will be able to Factor by Grouping terms When 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.

Factoring Four Term Polynomials Do now: find the GCf of the first two terms and the last two terms:

3x2 and 6 Group together and Factor each one separately: They share a common factor of (x-4)

Write 2 factors: Write the common factor once and put the outside terms together:

Factor by Grouping Example 1: FACTOR: 3xy - 21y + 5x – 35 Factor the first two terms: 3xy – 21y Factor the last two terms: + 5x - 35 =

Factor by Grouping Example 1: FACTOR: 3xy - 21y + 5x – 35 Factor 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 factor

Factor by Grouping Example 1: FACTOR: 3xy - 21y + 5x – 35 Factor 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 factor Now you have a common factor (x - 7) (3y + 5)

Factor by Grouping Example 2: FACTOR: 6mx – 4m + 3rx – 2r Factor the first two terms: 6mx – 4m = Factor the last two terms: + 3rx – 2r =

Factor by Grouping Example 2: FACTOR: 6mx – 4m + 3rx – 2r Factor 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 factor Now you have a common factor (3x - 2) (2m + r)

Factor by Grouping Example 3: FACTOR: y3– 5y2 -4y +20

Factor by Grouping Example 3: FACTOR: y3– 5y2 - 4y +20 Factor the first two terms: y3– 5y2 = y2 (y - 5) Factor the last two terms: - 4y +20 = -4 (y – 5) The green parentheses are the same! y2 (y - 5) and -4 (y - 5) Now you have the difference of two squares! look at red ( ): (y - 5) (y2 - 4) : answer: (y - 5) (y - 2) (y + 2)

See worksheet “Factor by grouping” Try first 4 problems.

So the solution set is { 5,2,-2} Using Factor by Grouping to solve a polynomial function: From the last example, suppose it was an equation….. y3– 5y2 - 4y +20 = 0 (y - 5) (y - 2) (y + 2) = 0 y=5 y = 2 y=-2 So the solution set is { 5,2,-2}

Factor first, then set factors = 0 X-4=0

solve X-4=0 X=4

Hand this one in: Solve for all roots: 3x3 - 4x2 -27x +36 = 0

Factoring Trinomials

Factoring Trinominals When trinomials have a degree of “2”, they are known as quadratics. We learned earlier to use the last term’s factors to factor trinomials that had a “1” in front of the squared term. x2 + 12x + 35 So… 7 and 5 or 35 and 1

Factoring Trinominals When trinomials have a degree of “2”, they are known as quadratics. We learned earlier to use the last term’s factors to factor trinomials that had a “1” in front of the squared term. x2 + 12x + 35 (x + 7)(x + 5) Because 7 + 5 = 12!

More Factoring Trinomials When there is a coefficient larger than “1” in front of the squared term, we can use a method we will call, the “am” add, multiply method to find the factors. Always remember to look for a GCF before you do ANY other factoring.

More Factoring Trinomials Let’s try this example 3x2 + 13x + 4 (3x )(x ) Write the factors of the last term…1,4 2,2 Multiply using foil until you get the middle term of the trinomial. If so, you’re done!

More Factoring Trinomials 3x2 + 13x + 4 (3x + 1 )(x + 4 ) 3x2 + 12x + 1x + 4 = 3x2 + 13x + 4 ✓

Difference of Squares

Difference of Squares When factoring using a difference of squares, look for the following three things: only 2 terms minus sign between them both terms must be perfect squares No common factors If all of the above are true, write two ( ), one with a + sign and one with a – sign : ( + ) ( - ).

Try These, (if possible) 1. a2 – 16 2. x2 – 25 3. 4y2 – 16 4. 9y2 – 25 5. 3r2 – 81 6. 2a2 + 16

answers: 1. a2 – 16 (a + 4) (a – 4) 2. x2 – 25 (x + 5) (x – 5) 3. 4y2 – 9 (2y + 3) (2y – 3) 4. 9y2 – 25 (3y + 5) (3y – 5) 5. 3r2 – 81 *3 is not a square! 6. a2 + 16 Not a difference!

Perfect Square Trinomials

Perfect Square Trinomials When factoring using perfect square trinomials, look for the following three things: 3 terms last term must be positive first and last terms must be perfect squares If all three of the above are true, write one ( )2 using the sign of the middle term.

Try These 1. a2 – 8a + 16 2. x2 + 10x + 25 3. 4y2 + 16y + 16

Factoring Completely

Factoring Completely Now 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 possible factors. Let’s review all the ways to factor.

Types of Factoring Look for GCF first. Count the number of terms: 4 terms – factor by grouping 3 terms - look for perfect square trinomial if not, try “am” method 2 terms - look for difference of squares If any ( ) still has an exponent of 2 or more, see if you can factor again.

These may take 2 steps! 1. 3r2 – 18r + 27 2. 2a2 + 8a - 8

Answers: 1. 3r2 – 30r + 27 3(r2 - 10r + 9) 3(r – 9) (r – 1) 2. 2a2 + 8a – 8 2(a2 + 4a – 4)

Solving Equations by Factoring Completely Do Now: Factor completely Solve for x

Steps to Solve Equations by Factoring Completely set each factor = 0 and solve for the unknown. x3 + 12x2 = 0 1. Factor GCF x2 (x + 12) = 0

Steps to Solve Equations by Factoring Completely set each factor = 0 and solve for the unknown. x3 + 12x2 = 0 1. Factor GCF x2 (x + 12) = 0 2. (set each factor = 0, & solve) x2 = 0 x + 12 = 0 x=0 x = -12 You now have 2 answers, x = 0 and x = -12

Factor completely:

Factor completely: X =

Solving higher degree functions Quadratic form: ax4 + bx2 + c = 0 Example: x4 +2x2 -24 = 0 Factor: (x2 )(x2 )=0

Solving higher degree functions Quadratic form: ax4 + bx2 + c = 0 Example: x4 +2x2 -24 = 0 Factor: (x2 +6 )(x2 – 4 ) = 0 x2 +6=0 x2 – 4 =0 x2 =-6 x2 = 4 x = 2, -2

Try this one: X4 – 13x2 +36 = 0

Factor first: X4 – 13x2 +36 = 0 (x2 – 9)(x2 – 4)=0

Solutions: X4 – 13x2 +36 = 0 (x2 – 9)(x2 – 4) X2-9=0 x2-4=0 This one can be verified on the calculator. X = 2,-2,3,-3

Hand this one in Quadratic form:

Ans: X=