How to Factor!

1) Greatest common factor Always work in the following order… 1) Greatest common factor 2) Perfect Trinomial Square 3) Quadratic functions 4) Difference of two Squares… 5) By grouping

1) Greatest common factor Look for… A constant, A variable, Or a combination of both. Ex 1: Factor 7x+7 …… Ex 2: Factor 3x+x² ….. Ex 3: Factor 2x² - 8x…

2) Perfect Trinomial Square x² + 8x + 16 or x² - 8x +16 Is the first term a perfect square? Is the last term a perfect square? Is the middle term… 2· =2 ·x ·4 ? If so, it factors as… Ex 1: x² +8x+ 16 = (x + 4)2 Use the middle sign Ex 2: x² -8x + 16 = (x - 4)2 Use the middle sign

2) Quadratic Functions Ex: x 2 + 10x + 16 Possible Products a) Middle term is positive Ex: x 2 + 10x + 16 Possible Products 16 = 1 x 16 =2 x 8 = 4 x 4 But Middle term must add up to +10! 2 +8=10 Factors as : (x + 2)(x + 8)

b) Middle term is negative Ex: x 2 - 10x + 16 Possible Products 16 = 1 x 16 = 2 x 8 = 4 x 4 But Middle term must add up to -10! -2 + -8= -10 Factors as : (x - 2)(x - 8)

That’s fine… BUT what if a≠1 Like …2x 2 - 5x – 12 ?

Bustin’ da middle term!! … Ex: 2x 2 - 5x – 12 Multiply (2)(-12) = -24 Possible Factors: -24 = -1 x 24 = -2 x 12 = 3 x 8 = -4 x 6 Yet… the Middle term is (- 5x) so, we need more negatives! 3x + - 8x and…

Bustin’ da Middle!! Cont… 2x 2 - 5x - 12 Original 2x 2 + 3x + -8x -12 Bust Middle. (2x 2 + 3x) + (-8x-12) Group. x(2x+3) – 4(2x+3) Factor GCF. (2x+3) (x-4) (Write what they have in common), then (write what’s left).

On your own… Yet…the middle term is(- 8x) ,so we need more negatives! Ex: 6x 2 - 8x – 8 Multiply (6)(-8) = -48 Possible Factors: -48 = -1 x 48 = -2 x 24 = -3 x16 = 4 x 12 = -6 x 8 Yet…the middle term is(- 8x) ,so we need more negatives! -12x + 4x so…

Bustin’ da Middle!! Cont… 6x 2 - 8x - 8 Original 6x 2 - 12x + 4x -8 Bust Middle. (6x 2 -12x) + (4x-8) Group. 6x(x-2) + 4(x-2) Factor GCF. (x-2) (6x+4) (Write what they have in common), then (write what’s left).

3a) Difference of two Squares x² - 16 Is the first term a perfect square? Is the second term a perfect square? Is there a minus between the two terms? Are there only two terms?...if yes Factor as: ( + )( – ) Ex: x 2 - 16 = (x+4)(x-4)

3b) Difference of 2 cubes a 3- b 3 = (a - b)(a 2+ ab + b 2) Factor: 64x 3 - 27 Factors as…

3c) Sum of 2 cubes a 3+ b 3 = (a + b)(a 2 - ab + b 2) Factor: 64x 3 + 27 Factors as…

3d) Difference of 4th powers is the “hidden” form of the difference of two squares a 4- b 4 = =(a 2 - b 2)(a 2+ b 2) Factor… Factors as… = (a - b)(a + b)(a 2+ b 2)

5) Grouping Used when there are 4 or more terms. Ex: 2ab + 14a + b + 7 (2ab +14a) + (b + 7) Group 2a(b+7) + 1(b+7) … factor, also 1 (b+7) (2a+1) (Write what they have in common), then (write what’s left).

Grouping cont…. Ex: xy + 3x – y²- 3y (xy + 3x) + (-y²-3y)… Always use a plus sign between parentheses, x(y+3) - y(y+3) then factor GCF (y+3) (x-y) (Write what they have in common), then (write what’s left).

Finding the zero’s (or solving) Try to factor… His implies that either expression must be 0 There are TWO solutions: x=6; x=-5

Something different… Solve by substitution.

Using Synthetic Division Factoring Using Synthetic Division

Synthetic Division We can use this method as long as you know one of the zero’s, Place the zero’s value of x here. 2 1 -4 5 -2 Multiply these and put answer above line in next column Multiply these and put answer above line in next column Multiply these and put answer above line in next column Bring first number down below line 2 Add these up - 4 2 Add these up Add these up 1 -2 1 This is the remainder List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0. Put variables back in (one x was divided out in the process, so the first term is x²). Is the factored form

You Try to factor! - 2 4 8 -25 -50 - 8 50 4 x2 + x - 25 set the factor = 0 and solve for x, this will be your divisor. - 2 4 8 -25 -50 Multiply these and put answer above line in next column Multiply these and put answer above line in next column Multiply these and put answer above line in next column Bring first number down below line - 8 Add these up 50 Add these up Add these up No remainder so x + 2 IS a factor because it divided in evenly 4 x2 + x - 25 So the answer is the divisor times the quotient: List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0. Put variables back in (one x was divided out in the process, so the first term is x²). You could factor even further…

You can use synthetic division to divide polynomials Set divisor = 0 and solve. Put answer here. x + 3 = 0 so x = - 3 1 - 3 1 6 8 -2 Multiply these and put answer above line in next column Multiply these and put answer above line in next column Multiply these and put answer above line in next column Bring first number down below line - 3 Add these up - 9 3 Add these up Add these up 1 x2 + x 3 - 1 1 This is the remainder List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0. Is the quotient Put variables back in (one x was divided out in the process, so the first term is x²).

CAVEAT…… Find the quotient 0 x3 0 x Set divisor = 0 and solve. Put answer here. x - 4 = 0 so x = 4 1 4 1 0 - 4 0 6 Multiply these and put answer above line in next column Multiply these and put answer above line in next column Multiply these and put answer above line in next column Multiply these and put answer above line in next column Bring first number down below line 4 Add these up 16 Add these up 48 Add these up 192 Add these up 1 x3 + x2 + x + 4 12 48 198 This is the remainder List all coefficients (numbers in front of x's) and the constant along the top. Don't forget the 0's for missing terms. Now put variables back in (remember one x was divided out in the process so first term is x3). Is the quotient