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Factoring Quadratic Expressions. Specific Expressions. Trinomial – Binomial – Monomial – . Consisting of three terms (Ex: 5x 3 – 9x 2 + 3). Consisting of 2 terms (Ex: 2x 6 + 2x). Consisting of one term (Ex: x 2 ). Quadratic Expression.
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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)
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: PRODUCT SUM
Factor 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:
The product of one diagonal always equals the product of the other diagonal. A Pattern with Generic Rectangles Example: 10x. -6x = -60x2 10x -15 4x2. -15 = -60x2 4x2 -6x
Factoring with the Box and Diamond Factor: cis 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 +6 3x 4x (2x2)(6) 12x2 2x2 GCF 2x 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 Factor: Product c (3x2)(-10) -30x2 5 -10 15x -2x ax2c -2x 15x 3x2 GCF x ___ bx ax2 13x 3x -2 Sum ( x + 5 )( 3x – 2 )
Factoring: Different Order Factor: Rewrite in Standard Form: ax2 + bx + c Product c (15x2)(-77) -1155x2 11 -77 33x -35x ax2c -35x 33x 15x2 GCF 5x ___ bx ax2 -2x 3x -7 Sum ( 5x + 11 )( 3x – 7 )
Factoring: Perfect Square Factor: Product c (x2)(9) 9x2 3 9 3x 3x ax2c 3x 3x x2 GCF x ___ bx ax2 6x x 3 Sum ( x + 3 )( x + 3 ) ( x + 3 )2
Factoring: Missing Terms Factor: Product c (9x2)(-4) -36x2 2 -4 6x -6x ax2c -6x 6x 9x2 GCF 3x ___ bx ax2 0 3x -2 Sum ( 3x + 2 )( 3x – 2 )
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: x2 ÷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 )( 2x – 1 ) Don’t forget the GCF Product c (2x2)(-3) -6x2 3 -3 6x -x ax2c -x 6x 2x2 GCF x ___ bx ax2 5x 2x -1 Sum
Factoring: Factoring Completely Factor: Ignore the GCF and factor the quadratic Reverse Box to factor out the GCF 3x(x + 3)(x – 5) Don’t forget the GCF Product c (x2)(-15) -15x2 3 -15 3x -5x ax2c -5x 3x x2 GCF x ___ bx ax2 -2x x -5 Sum
Factoring: Forgot to Factor a GCF Factor: No, it is not factored completely because one of the factors still has a GCF bigger than 1. When factoring the above expression a student came up with the following answer. Is it factored completely? 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 When the x2 term is negative, it is difficult to factor. Factor: Ignore the GCF and factor the quadratic Reverse Box to factor out the negative -( x + 6 )( x + 7 ) Don’t forget the GCF Product c (x2)(42) 42x2 6 42 6x 7x ax2c 6x 7x x2 GCF x ___ bx ax2 13x x 7 Sum