Factoring The simple Case CCM2 11.1-4

What is factoring? Factoring undoes the multiplication of polynomials Polynomials that cannot be reduced further into more basic factors are said to be prime. This is similar to prime numbers. A polynomial that is has been factored is said to be in factored form. For example, 4x (7x + 3) (x - 5) is in factored form. The degree of a polynomial in factored form is the sum of the degrees of each factor. For example, 4x (7x + 3) (x - 5) has a degree of 3.

Types of Factoring Greatest Common Factor Simple Factoring Grouping All terms have a given number and/or variables as a factor Simple Factoring Trinomials with a leading coefficient is 1 Grouping Trinomial with a Leading Coefficient that is not 1, or A polynomial with four terms Special Cases Difference of Squares Perfect Squares

Greatest Common Factor Pull out the GCF Numbers: for the coefficients and constant this just like the greatest common factor you learned in 5th grade Variables, use the lowest power of each variable that is common to all terms Next to the GCF write what results when you divide the polynomial by the GCF in parenthesis Example: =

Quadratics in Standard form (Revisited) The Standard form of any quadratic is ax2 +bx + c, where a, b and c are real numbers. The leading coefficient is the coefficient in the highest degree term. Example: What are the coefficients of 3x2 + 4x – 5? 3, 4 What is the leading coefficient? 3

Factoring with a leading coefficient of 1 Find all factor pairs of c ( f1 ∙ f2 = c) Select the pair that adds up to b ( f1+ f2 = b) The result is ( x + f1 )( x + f2 ) Example: x2 + 5x + 6 Factors of 6 = 1x6, 2x3 Note that 2 + 3 = 5 So, the result is ( x + 2 )( x +3 )

Dealing with signs The sign of c: + means the two factors have the same sign - means the two factors have opposite sign The sign of b is the sign of the larger maginitude factor. Example: x2 + 5x - 6 Factors of -6 = -1 ∙ 6, -2 ∙ 3. Note that -1 + 6 = 5 So, the result is ( x - 1 )( x + 6 )

Using the TI 83/84 to factor c Press [ Y= ] Type in Y1= “c”/X Press [ 2nd ] [Graph] for the Table Scroll down from 1 and note whole number pairs Factor x2 + 76x + 288 Y1= 288/X [ 2nd ] [Graph] gives 1x288, 2x144, 3x96, 4x72, 6x48,8x36, etc. …. Note that 4 + 72 = 76 Solution is (x+4)(x+72)

Summary What is factoring? Prime Factors Greatest Common Factors Simple Factoring Dealing with Signs when factoring. Thank You