How to Factor Binomials

Factoring Checklist I. Binomials II. Trinomials GCF Difference of squares Cubes II. Trinomials Trial & Error Circle, slide, divide

1) GCF: Greatest common factor It may be a constant, a variable, of a combination of both (3, X, 4X). Simply find what each term has in common and “take it out”. Remember, you should always be able to multiply it back together to get what you started with. Ex 1: x2 + 7x = (x + 7) x Ex 2: 4x3 + 2x2 - 8x = 2x (2x2 + x - 4)

2) Difference of Squares Another type of problem that you may encounter would be something like this. x2 – 25

2) Difference of Squares x2 – 25 Is the first term a perfect square? What are your perfect squares? 1 4 9 16 25 36 And so on… you need to know up to 132 = 169 How do you decide if an exponent is a perfect square or not? It has to have an even exponent.

2) Difference of Squares x2 – 25 Is the first term a perfect square? Is the second term a perfect square? Is there a minus between the two terms? If the answer to all three of these questions is yes, then it’s a difference of squares. Difference of squares factor into conjugates.

So for this problem: x2 – 25 It factors to be: (x + 5)(x – 5)

Problem #2: x2 – 49 (x + 7)(x – 7)

Problem #3: x2 + 16 Cannot be factored

Problem #4: 25x6 – 64y8 (5x3 + 8y4)(5x3 – 8y4)

Problem #5: x4 – 16 (x2 + 4)(x2 – 4) (x2 + 4)(x – 2)(x + 2)

Problem #6: 4x2 – 16 4(x2 – 4) 4(x – 2)(x + 2)

Difference of squares practice 1) 2) 3) 4) You cannot factor this. It is a SUM of squares, not a difference. So, for the answer, write “PRIME” or “does not factor”.

Cubes x3 + 8 = Be sure to check for GCF and Diff of Squares. Then decide if it’s cubes. What are your cubes? 1 8 27 64 125 (this is far enough) How do you know if variables are cubes? Their exponents can be divided by three!

Cubes x3 + 8 falls under the “cubes” category. To factor, cubes always factor into a binomial times a trinomial. x3 + 8 = (__ __) (__ __ __) To fill in the first two blanks, take the cube root of the two terms. x3 + 8 = (x 2) (__ __ __)

Cubes x3 + 8 = (x 2) (__ __ __) To fill in the next three blanks, Square the 1st to get the 1st Square the last to get the last Multiply them together to get the middle x3 + 8 = (x 2) (x2 2x 4)

Cubes x3 + 8 = (x 2) (x2 2x 4) Now the only thing left are the signs. To get the signs, remember SOP! S- Same O- Opposite P- Positive x3 + 8 = (x + 2) (x2 - 2x + 4)

( )( ) So, let’s see that same problem again. x + 2 x2 – 2x + 4 ( )( ) x + 2 x2 – 2x + 4 x3 + 8 = Since it’s cubes, it factors to be a binomial times a trinomial. Take the cube roots to get both of the terms for the first parentheses. Use those two terms for the second parentheses: Square the first to get the first. Square the last to get the last. Multiply them together to get the middle. Then use SOP for the signs.

Problem #2: x3 + 27 (x + 3)(x2 – 3x + 9)

Problem #3: 8x3 – 125 (2x – 5)(4x2 + 10x + 25)

Problem #4: 27x9 – 8y21 (3x3 – 2y7)(9x6 + 6x3y7+ 4y14)

Cubes practice 1) 2) 3)

(x – 1)(x2 + x + 1) (x + 1)(x2 – x + 1) Difficult Problem: x6 – 1 Difference of squares first! (x3 – 1)(x3 + 1) Now, both of these parentheses are cubes! (x – 1)(x2 + x + 1) (x + 1)(x2 – x + 1)

Warm Up #2

How to Factor Trinomials

Factoring Checklist I. Binomials II. Trinomials GCF Difference of squares Cubes II. Trinomials Trial & Error Circle, slide, divide

1) GCF: Greatest common factor Don’t forget…when you look for GCF, you are undoing the distributive property. 1) 4x3 + 2x2 - 8x = 2x(2x2 + x - 4) 2) 6x3y4 + 3x2y - 12x4y5 = 3x2y (2xy3 + 1 – 4x2y4)

Before we go to “circle, slide, divide… What two numbers multiply to give you 20 that add to give you 9? 4 x 5 Another way to ask the same question is if they ask, “What are the factors of 20 that add to give you 9?” Let’s try another… What are the factors of 24 that add to give you -11? -8 x -3 One more try… What are the factors of -12 that add to give you -1? -4 x 3

2) Circle, slide, divide (__ __)(__ __) Check for GCF. Then do trial and error. The factoring will be two binomials, so draw 2 parentheses. Ex. A If the leading coefficient is a one, it is very easy to do. (__ __)(__ __) Find two factors of -6 that add to get the middle term. Don’t forget, you can multiply your parentheses together to see if you’re right.

Now, let’s look and see why we couldn’t have used 2 and 3 instead of 1 and 6. Ex. A (__ __)(__ __) Notice, you don’t get what you started with.

Ex. B Check for GCF. Then do trial and error. The factoring will be two binomials, so draw 2 parentheses. Find two factors of 15 that add to get 8. (__ __)(__ __) Now multiply your parentheses together to see if you’re right.

(__ __)(__ __) Ex. C Let’s try 6 and 4. (__ __)(__ __) This is why 6 and 4 don’t work. What else can we try? We need factors of -24 that add to give me -10.

(__ __)(__ __) Ex. C Let’s try 2 and -12. (__ __)(__ __) Hooray! This is the correct answer!

4( )( ) Ex. D First, TAKE OUT THE GCF!!! Then, factor what’s leftover. Multiply to see if you are correct. 4( )( ) The most factored out part is the final answer!

Because the leading coefficient is NOT a one, we need to “circle, slide, divide”. Ex. E Multiply the front number times the back number. Find two things that multiply to get the last term that also SUBTRACTS to get the middle. ( )( ) Now, divide by the original leading coefficient. If it reduces, do it. If there is still a denominator, it needs to be placed in front of the x in that parentheses for the final answer.

(__ __)(__ __) (__ __)(__ __) (__ __)(__ __) When the leading coefficient is not one, we will use the circle, slide divide method of factoring. Ex. F Circle the leading coefficient Slide it over and multiply it by the last term. Now, find the factors of 72 that add to be -17 (__ __)(__ __) Now, divide your factors by the circled term… and simplify the fractions!!!! (__ __)(__ __) (__ __)(__ __) Finally, Move the simplified denominators to the front…

8 (__ __)(__ __) (__ __)(__ __) When the leading coefficient is not one, we will use the circle, slide divide method of factoring. Ex. G Circle the leading coefficient Slide it over and multiply it by the last term. Now, find the factors of -96 that add to be -29 (__ __)(__ __) Now, divide your factors by the circled term… and simplify the fractions!!!! (__ __)(__ __) 8 Finally, Move the simplified denominators to the front…

3 4 (__ __)(__ __) (__ __)(__ __) When the leading coefficient is not one, we will use the circle, slide divide method of factoring. Ex. H Circle the leading coefficient Slide it over and multiply it by the last term. Now, find the factors of -96 that add to be -29 (__ __)(__ __) Now, divide your factors by the circled term… and simplify the fractions!!!! (__ __)(__ __) 3 4 Finally, Move the simplified denominators to the front…

Here’s some practice for you: 1) 2) 3) 4)

Practice: Factor completely.

Answers to Practice:

Find your groups and sit with them now. 2nd Block Group A Kendal A Jake M Hayden M Group B Steven A Tat M Summer N Carl W Group C Mitchell B Michael L Josh P Jordan U Group D Andrew B Ben K Chris P Jacob T Group E Dalton C Adam J Abbi R Jessica S Group F Antonio D Tay H Frederick R Brynn M Group G Keevan G Chesnee S Cody S Group H Ella E Emma E Ross S Kelsey S

Find your groups and sit with them now. 4th Block Group A Kory C Mekavia L Makala L Keyna W Group B Kesha C Cait K Nick L Korea W Group C Deanna C Spencer J Joey D Shay W Group D Erin C Destiny H Celeste M Addy W Group E Cameron C Kindle N Troy H Elliot S Group F Kat D Zac H Steve O Group G Jermesha F Kristina F Shiesha O Kyanna S

Practice #2: Factor completely.

Answers to Practice #2:

A little more practice before moving to 4-terms...

FACTORING 4-TERM POLYNOMIALS

Factoring Checklist I. Binomials II. Trinomials GCF Difference of squares Cubes II. Trinomials Circle, Slide, Divide III. 4-term polynomials Grouping Compound squares

Factor 12ac + 21ad + 8bc + 14bd Do you have a GCF for all 4 terms? No Group the first 2 terms and the last 2 terms. (12ac + 21ad) + (8bc + 14bd) Find the GCF of each group. 3a (4c + 7d) + 2b(4c + 7d) The parentheses are the same! (3a + 2b)(4c + 7d)

Factor: rx + 2ry + kx + 2ky Check for a GCF: None You have 4 terms - try factoring by grouping. (rx + 2ry) + (kx + 2ky) Find the GCF of each group. r(x + 2y) + k(x + 2y) The parentheses are the same! (r + k)(x + 2y)

Factor: 2x2 - 3xz - 2xy + 3yz Check for a GCF: None Factor by grouping. Keep a + between the groups! (2x2 - 3xz) + (- 2xy + 3yz) Find the GCF of each group. x(2x - 3z) + y(- 2x + 3z) The signs are opposite in the parentheses! Additive Inverses – Use the -1 factor! x(2x - 3z) - y(2x - 3z) (x - y)(2x - 3z)

Factor: 16k3 - 4k2p2 - 36k + 9p2 Check for a GCF: None Factor by grouping. Keep a + between the groups! (16k3 - 4k2p2 ) + (-36k + 9p2) Find the GCF of each group. 4k2(4k - p2) + 9(-4k + p2) The signs are opposite in the parentheses! Additive Inverses – Use the -1 factor! 4k2(4k - p2) - 9(4k - p2) (4k2 - 9)(4k - p2) (2k - 3)(2k + 3)(4k - p2)

Practice: Factor completely. 1) ax + bx + ay + by 2) 3p – 6d + mp – 2md 3) 4xy – 3xz – 8y + 6z 4) 24ac + 30 bc – 4ab –5b2 5) x2y + 3x2 – 4y – 12 1) (x + y) (a + b) 2) (3 + m) (p – 2d) 3) (x – 2) (4y – 3z) 4) (4a + 5b) (6c – b) 5) (x – 2) (x + 2) (y + 3)

Homework: Factor completely. 1) wx + wy – xz – yz 2) 2ax + 6a – 5x2 – 15x 3) 6xy – 8x + 9y – 12 4) 4ax + 6x2y – 12x + 16x3 5) x3 + 3x2 – 25x - 75 1) (x + y) (w – z) 2) (x + 3) (2a – 5x) 3) (3y – 4) (2x + 3) 4) 2x(2a + 3xy – 6 + 8x2) 5) (x – 5)(x + 5)(x + 3)

Homework: Factor completely. 1) wx + wy – xz – yz w (x + y) + z (-x – y) w (x + y) – z (x + y) (x + y) (w – z)

Homework: Factor completely. 2) 2ax + 6a – 5x2 – 15x (2ax + 6a) + (– 5x2 – 15x) 2a(x + 3) + 5x(-x – 3) 2a(x + 3) – 5x(x + 3) (x + 3) (2a – 5x)

Homework: Factor completely. 3) 6xy – 8x + 9y – 12 2x (3y – 4) + 3 (3y – 4) (3y – 4) (2x + 3)

Homework: Factor completely. 4) 4ax + 6x2y – 12x + 16x3 2x (2a + 3xy – 6 + 8x2)

Homework: Factor completely. 5) x3 + 3x2 – 25x - 75 x2(x + 3) – 25(x + 3) (x2 - 25)(x + 3) (x – 5)(x + 5)(x + 3)

1) (x + y) (w – z) 2) (x + 3) (2a – 5x) 3) (3y – 4) (2x + 3) 4) 2x(2a + 3xy – 6 + 8x2) 5) (x – 5)(x + 5)(x + 3) On your worksheet, omit problems 22 and 24.