Factoring Quadratic Expressions Specific Expressions Trinomial Consisting of

Factoring Quadratic Expressions

Specific Expressions • Trinomial – Consisting of three terms 3 2 (Ex: 5 x – 9 x + 3) • Binomial – Consisting of 2 terms 6 (Ex: 2 x + 2 x) • Monomial – Consisting of one term 2 (Ex: x )

Quadratic Expression An expression in x that can be written in the standard form: 2 ax + bx +c Where a, b, and c are any number except a ≠ 0.

Factoring The process of rewriting a mathematical expression involving a sum to a product. It is the opposite of distributing. Example: SUM PRODUCT

Factor 2 If x + 8 x + 15 = ( x + 3 )( x + 5 ) then x + 3 and x + 5 are called factors of x 2 + 8 x + 15 (Remember that 3 and 4 are factors of 12 since 3. 4=12)

Finding the Dimensions of a Generic Rectangle Mr. Wells’ Way to find the product for a generic rectangle: Make sure to Check First, find the POSITIVE Greatest Common Factor of two terms in the bottom row. 5 10 x -15 2 x 2 -6 x 4 x 2 x Lastly, write the answer as a Product: -3 Second, find missing WHOLE NUMBER dimensions on the individual boxes.

A Pattern with Generic Rectangles The product of one diagonal always equals the product of the other diagonal. Example: 10 x 4 x 2 -15 -6 x 10 x. -6 x = -60 x 2 4 x 2. -15 = -60 x 2

Factoring with the Box and Diamond Fill in the results from the diamond and find the dimensions of the box: 3 GCF 2 x ___ ax 2 is always in the bottom left corner 3 x +6 2 x 2 4 x x 2 c is always in the top right corner Because of our pattern, the missing boxes need to multiply to: 2 Diamond Problem Factor: Write the expression as a product: ( 2 x + 3 )( x + 2 ) (2 x )(6) 12 x 2 4 x 3 x 7 x The missing boxes also have to add up to bx in the sum

Factoring Example Factor: Product 5 GCF ___ x c 15 x 3 x 2 ax 2 3 x -10 -2 x (3 x 2)(-10) -30 x 2 -2 x -2 ( x + 5 )( 3 x – 2 ) ax 2 c bx 13 x Sum 15 x

Factoring: Different Order Factor: Rewrite in Standard Form: ax 2 + bx + c Product c 11 33 x 2 15 x GCF 5 x ___ ax 2 3 x -77 -35 x (15 x 2)(-77) -1155 x 2 -35 x -7 ( 5 x + 11 )( 3 x – 7 ) ax 2 c bx -2 x Sum 33 x

Factoring: Perfect Square Factor: Product (x 2)(9) c 3 GCF ___ x 3 x x 2 ax 2 x 9 9 x 2 3 x 3 x ax 2 c bx 6 x 3 Sum ( x + 3 )2 3 x

Factoring: Missing Terms Factor: Product c 2 GCF 3 x ___ 6 x 9 x 2 ax 2 3 x -4 -6 x (9 x 2)(-4) -36 x 2 -6 x -2 ( 3 x + 2 )( 3 x – 2 ) ax 2 c bx 0 Sum 6 x

Factoring: Which Expression is correct? Notice that every term is divisible by 2 Factor: If you use the box and diamond, the following products are possible: x 2 ÷ 2 Which is the best possible answer?

Factoring: Factoring Completely Factor: Ignore the GCF and factor the quadratic Reverse Box to factor out the GCF 5( x + 3 )( 2 x – 1 ) Product (2 x 2)(-3) c 3 GCF ___ x 6 x 2 x 2 ax 2 2 x -3 -x -1 Don’t forget the GCF -6 x 2 -x ax 2 c bx 5 x Sum 6 x

Factoring: Factoring Completely Factor: Ignore the GCF and factor the quadratic Reverse Box to factor out the GCF 3 x(x + 3)(x – 5) Product c 3 GCF ___ x 3 x x 2 ax 2 x -15 -5 x -5 (x 2)(-15) Don’t forget the GCF -15 x 2 -5 x ax 2 c bx -2 x Sum 3 x

Factoring: Forgot to Factor a GCF Factor: When factoring the above expression a student came up with the following answer. Is it factored completely? No, it is not factored completely because one of the factors still has a GCF bigger than 1. Factor out the GCF with a reverse box Substitute the result: NOTE: I do not recommend relying on this. It can be used IF you forget to check for a GCF.

Factoring: Just Factoring a GCF Factor: Reverse Box to factor out the GCF There is no longer a quadratic, it is not possible to factor anymore. There is not always more factoring after the GCF.

Factoring: Ensuring “a” is Positive Factor: Reverse Box to factor out the negative When the x 2 term is negative, it is difficult to factor. Ignore the GCF and factor the quadratic -( x + 6 )( x + 7 ) Product (x 2)(42) c 6 GCF ___ x 6 x x 2 ax 2 x 42 7 x 7 Don’t forget the GCF 42 x 2 6 x ax 2 c bx 13 x Sum 7 x