5.1 Factoring a monomial from a polynomial Factors-numbers (or variables) that you multiply together to get a product The factors of 30 = 1,2,3,5,6,10,15,30 To factor something is the opposite of multiplying Multiplying 3(x+2) = 3x + 6 Factoring 3x + 6 = 3(x+2)

5.1 Factoring a monomial from a polynomial GCF Greatest Common Factor a number that will divide evenly into two or more numbers; the largest one you can find. Find GCF of 48 and 60 is 12 Factors of 48=1,2,3,4,6,8,12,16,24,48 Factors of 60=1,2,3,4,5,6,10,12,15,20,30,60

5.1 Factoring a monomial from a polynomial We don’t want to have to list all the factors every time we need to find a gcf. Sometimes we can tell by looking at the numbers. GCF of 12 and 16 is 4 If not use the prime factorization method: Make a factor tree for each number. Once you have all primes, put your prime factorization into exponent form. Compare the two prime factorizations and see what they have in common. (Do 48 and 60 on board) Now find the GCF of 18 and 24

5.1 Factoring a monomial from a polynomial You can apply this same idea to terms: GCF of 18y2, 15y3, and 27y5 Is 3y2 GCF of 2(x+y) and 3x(x+y) Is (x+y)

5.1 Factoring a monomial from a polynomial To factor a monomial from a polynomial, 1)Find GCF of all the terms 2) Put GCF out front 3) Use the dist property in reverse to factor out the GCF from EACH term If you multiply it back in, you should get what you started with

5.1 Factoring a monomial from a polynomial 15x – 20 = 5 (3x – 4) 6y2 + 9y5 = 3y2 (2 + 3y3) 35x2 – 25x + 5 = 5 (7x2 - 5x +1) 2x(x - 3) – 5(x - 3) = (x - 3) (2x - 5)

5.3 Factoring Trinomials This is getting ahead of us, but so you will understand where this fits in— These are the steps we use to factor something completely 1) Check for a GCF (you learned in 5.1) 2) Count the terms trinomial-FOIL backwards (5.3 and 5.4) binomial-DOS or sum/diff of cubes 4 terms-grouping

X2 + 7x + 12 GCF? No trinomial? Yes FOIL backwards: X2 + 7x + 12 = ( )( ) FOIL start with F – to get X2 you need x,x X2 + 7x + 12 = (x )(x ) FOIL now do L – to get 12, you could use any two factors of 12 (1,12 or 2,6 or 3,4) which pair will add or subtract to 7? 1,12 = 13 or 11 2,6 = 8 or 4 3,4 = 7 or 1

So 3,4 will work X2 + 7x + 12 = (x 3)(x 4) Now consider the signs: If the constant (12) is positive, both factors will have the same sign-both (+) or both (-) If the constant (12) is negative, the two factors will have different signs; one positive and one negative In this case, in order to get a (+)12, we need both signs to be the same. In addition, we want a (+) 7 in the middle so we would choose both positives X2 + 7x + 12 = (x + 3)(x + 4)

X2 + 7x + 12 = (x + 3)(x + 4) Lastly, check your smile lines Assuming you chose wisely, you shouldn’t need to check x times x. it should equal x2; You shouldn’t need to check 3 times 4. it should equal 12. But double check the middle term Can you see the 3x and 4x that will combine for a 7x in the middle?

5.3 Factoring Trinomials Factor X2 + x - 6 = (x + 3)(x - 2) start with F then L; consider 1,6 or 2,3 which will give you a 1 in the middle? 2,3 if we subtract them since the constant is negative, we want our signs to be a plus and a minus

5.3 Factoring Trinomials Factor X2 + 2x - 24 = (x + 6)(x - 4) F: x and x L: 1,24 or 2,12 or 3,8 or 4,6 Signs: -24 means you need a (+) (-) Arrange them so the two will be (+)

Trinomial signs binomial signs ax2 + bx + c ( + )( + ) ax2 - bx + c ( - )( - ) ax2 - bx - c ( + )( - ) and OR ax2 + bx - c ( - )( + )

5.3 Factoring Trinomials Factor 2X2 + 2x - 12 Check for GCF first ALWAYS!!! GCF=2 so 2(X2 + x – 6) F: x and x L: 1,2 or 2,3 Signs: -6 means you need a (+) (-) Arrange them so the one will be (+) 2X2 + 2x - 12 2(X2 + x – 6) 2(x + 3)(x - 2)

5.4 Factoring Trinomials when a=1 The same method applies here as in 5.3 but you will notice that the coefficient on the x2 term will impact our middle term so finding the pair of factors becomes more challenging. You will often have to do more trial and error on paper to find the set that works. There are two approaches: Guess and Check which follows immediately Grouping which comes after guess/check

Guess and Check 2x2 + 13x + 15 = start with F, then L, then signs, check MT (2x + 15)(1x + 1) 15x + 2x =17x NO (2x + 1)(1x + 15) 1x + 30x = 31x NO (2x + 5)(1x + 3) 5x + 6x = 11x NO (2x + 3)(1x + 5) 3x + 10x = 13x YES!

2x2 + 7x + 6 = for 6, consider 1,6 and 2,3 (2x + 6)(1x + 1) (2x + 1)(1x + 6) (2x + 2)(1x + 3) (2x + 3)(1x + 2) 4x + 3x = 7x YES!

3x2 + 20x + 12 = for 12, consider 1,12 and 2,6 and 3,4

5x2 - 7x - 6 = for 6, consider 1,6 and 2,3 (5x 1)(1x 6) (5x 6)(1x 1) (5x 2)(1x 3) (5x + 3)(1x - 2)

8x2 + 33x + 4 = for 4, consider 1,4 and 2,2 for 8, consider 1,8 and 2,4 (8x + 4)(1x + 1) (8x + 1)(1x + 4) (8x + 2)(1x + 2) (2x + 1)(4x + 4) (2x + 4)(4x + 1) (2x + 2)(4x + 2)

Grouping 2x2 + 13x + 15 where a=2 b=13 and c=15 Take a times c so 2(15) = 30 1) Find two numbers whose product is ac and whose sum is b In this case, two numbers whose product is 30 and whose sum is 13 Consider the factors of 30: 1,30 and 2,15 and 3,10, and 5,6 3,10 add to 13

2) Rewrite the middle term as the sum or difference of the two factors 2x2 + 13x + 15 2x2 + 10x + 3x + 15 3) Factor the new expression by grouping two and two GCF of the 1st two? 2x of the 2nd two? 3 2x (x+5) + 3 (x+5) Pull the common factor to the front now (x+5) (2x+3) DONE!

2x2 + 7x + 6 ac = 2(6) = 12 and b = 7 1,12 and 2,6 and 3,4 2x2 + 3x + 4x + 6 group x(2x + 3) + 2(2x + 3) Pull 2x+3 to the front (2x + 3)(x + 2)

3x2 + 20x + 12 ac = 3(12) = 36 and b = 20 1,36 and 2,18 and 3,12, and 4,9 and 6,6 3x2 + 18x + 2x + 12 group 3x(x + 6) + 2(x + 6) Pull to the front (x + 6)(3x + 2)

5x2 - 7x - 6 ac = 5(-6) = -30 and b = -7 1,30 and 2,15 and 3,10 and 5,6 Consider the signs here 3 and -10 = -7 5x2 - 10x + 3x - 6 group 5x(x - 2) + 3(x - 2) Pull to the front (x - 2)(5x + 3)

8x2 + 33x + 4 ac = 8(4) = 32 and b = 33 1,32 and 2,16 and 4,8 8x2 + 32x + 1x + 4 group 8x(x + 4) + 1(x + 4) Pull to the front (x + 4)(8x + 1)

5.5 Special formulas Recall the factoring process 1) Check for a GCF (you learned in 5.1) 2) Count the terms trinomial-FOIL backwards (5.3 and 5.4) binomial-DOS or sum/diff of cubes(5.5) 4 terms-grouping

Difference of squares DOS always involves perfect square numbers: like 1,4,9,16, 25, 36, 49, 64, etc. It is always “something squared minus something squared” a2 – b2 = (a + b)(a - b) x2 – 16 = x2 – 42 = (x + 4)(x - 4) 25x2 – 4 = (5x)2 –(2)2 = (5x + 2)(5x - 2) 36x2 – 49y2 = (6x)2 –(7y)2=(6x + 7y)(6x – 7y) “sum of squares” are always prime.

Difference of cubes/Sum of cubes DOC/SOC always involves perfect cube numbers: like 1, 8, 27, 64, 125, 216, etc. It is always “something cubed plus/minus something cubed” a3 – b3 = (a - b)(a2 + ab + b2) a3 + b3 = (a + b)(a2 - ab + b2)

Difference of cubes a3 – b3 = (a - b)(a2 + ab + b2) y3 – 125 = y3 –53 = (y - 5)(y2 + y(5) + 52) (y - 5)(y2 + 5y + 25)

Sum of cubes a3 + b3 = (a + b)(a2 - ab + b2) 8p3 + k3 = (2p)3 +(k)3 = (2p + k)((2p)2 - (2p)(k) + k2) (2p + k)(4p2 - 2pk + k2)