Factoring a Quadratic Expression

*Make sure to leave the right form on any assessment* Expression v. Equation Expression: Numbers, symbols, and operators grouped together that show the value of something. For example: 3x – 5 Equation: A Statement formed when a equality symbol is placed between two equal expressions. For example: y = 3x – 5 *Make sure to leave the right form on any assessment*

Specific Expressions Trinomial – Binomial – Monomial – Consisting of three terms (Ex: 5x3 – 9x2 + 3) Consisting of 2 terms (Ex: 2x6 + 2x) Consisting of one term (Ex: x2)

ax2 + bx +c Quadratic Expression An expression in x that can be written in the standard form: ax2 + 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

(Remember that 3 and 4 are factors of 12 since 3.4=12) If x2 + 8x + 15 = ( x + 3 )( x + 5 ) then x + 3 and x + 5 are called factors of x2 + 8x + 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 Second, find missing WHOLE NUMBER dimensions on the individual boxes. First, find the POSITIVE Greatest Common Factor of two terms in the bottom row. 5 10x -15 2x 4x2 -6x 2x -3 Lastly, write the answer as a Product:

Factoring with the Box and Diamond c is always in the top right corner Because of our pattern, the missing boxes need to multiply to: Fill in the results from the diamond and find the dimensions of the box: 3 3x 4x +6 (2x2)(6) 12x2 GCF 2x 2x2 4x 3x ___ Diamond Problem 7x x 2 ax2 is always in the bottom left corner Write the expression as a product: The missing boxes also have to add up to bx in the sum ( 2x + 3 )( x + 2 )

Factoring Example 1 Factor: (3x2)(-10) -30x2 5 15x -2x -10 -2x 15x GCF Product c (3x2)(-10) -30x2 5 15x -2x -10 ax2c -2x 15x GCF x 3x2 ___ bx ax2 13x 3x -2 Sum ( x + 5 )( 3x – 2 )

Factoring Example 2 Factor: (15x2)(-77) -1155x2 11 33x -35x -77 Rewrite in Standard Form: ax2 + bx + c Product c (15x2)(-77) -1155x2 11 33x -35x -77 ax2c -35x 33x GCF 5x 15x2 ___ bx ax2 -2x 3x -7 Sum ( 5x + 11 )( 3x – 7 )

Factoring Example 3 Factor: (x2)(9) 9x2 3 3x 9 3x 3x GCF x x2 6x x 3 Product c (x2)(9) 9x2 3 3x 9 ax2c 3x 3x GCF x x2 ___ bx ax2 6x x 3 Sum ( x + 3 )( x + 3 ) ( x + 3 )2

Factoring Example 4 Factor: (9x2)(-4) -36x2 2 6x -6x -4 -6x 6x GCF 3x Product c (9x2)(-4) -36x2 2 6x -6x -4 ax2c -6x 6x GCF 3x 9x2 ___ bx ax2 3x -2 Sum ( 3x + 2 )( 3x – 2 )

Factoring: Which Expression is correct? Notice that every term is divisible by 5 Factor: If you use the box and diamond, the following products are possible: x5 ÷5 Which is the best possible answer?

Factoring Completely Example 1 Ignore the GCF and factor the quadratic Reverse Box to factor out the GCF 5( x + 3 )( 2x – 1 ) Product Don’t forget the GCF c (2x2)(-3) -6x2 3 6x -x -3 ax2c -x 6x GCF x 2x2 ___ bx ax2 5x 2x -1 Sum

Factoring Completely Example 2 Ignore the GCF and factor the quadratic Reverse Box to factor out the GCF 3x(x + 3)(x – 5) Product Don’t forget the GCF c (x2)(-15) -15x2 3 3x -5x -15 ax2c -5x 3x GCF x x2 ___ bx ax2 -2x x -5 Sum

Factoring Completely Example 3 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 Completely Example 4 When the x2 term is negative, it is difficult to factor. Reverse Box to factor out the negative Ignore the GCF and factor the quadratic -( x + 6 )( x + 7 ) Product Don’t forget the GCF c (x2)(42) 42x2 6 6x 7x 42 ax2c 6x 7x GCF x x2 ___ bx ax2 13x x 7 Sum