MAT 105 FALL 2008 Review of Factoring and Algebraic Fractions Chapter 6 Review of Factoring and Algebraic Fractions

Section 6.2: Factoring: Common Factors and Difference of Squares MAT 105 FALL 2008 Section 6.2: Factoring: Common Factors and Difference of Squares Factoring is the reverse of multiplying. A polynomial or a factor is called _________________ if it contains no factors other than 1 (or -1) and itself.

THE FIRST STEP: Factoring Out the Greatest Common Monomial Factor MAT 105 FALL 2008 THE FIRST STEP: Factoring Out the Greatest Common Monomial Factor

Factoring the Difference of Perfect Squares MAT 105 FALL 2008 Factoring the Difference of Perfect Squares Recall: Difference of Squares:

Factoring the Difference of Perfect Squares MAT 105 FALL 2008 Factoring the Difference of Perfect Squares

Factor Completely: HINT: Always check for a GCF first!! MAT 105 FALL 2008 Factor Completely: HINT: Always check for a GCF first!!

MAT 105 FALL 2008 Factoring by Grouping (Consider grouping method if polynomial has 4 terms) Always start by checking for a GCF of all 4 terms. After you factor out the GCF or if the polynomial does not have a GCF other than 1, check if the remaining 4-term polynomial can be factored by grouping. Determine if you can pair up the terms in such a way that each pair has its own common factor. If so, factor out the common factor from each pair. If the resulting terms have a common binomial factor, factor it out.

MAT 105 FALL 2008 Factor Completely

MAT 105 FALL 2008 Factor Completely

I. Factoring Trinomials in the Form MAT 105 FALL 2008 Section 6.3: Factoring Trinomials I. Factoring Trinomials in the Form Recall: F L O + I To factor a trinomial is to reverse the multiplication process (UnFOIL)

Before you attempt to Un-FOIL MAT 105 FALL 2008 Before you attempt to Un-FOIL 1) Always factor out the GCF first, if possible. 2) Write terms in descending order. Now we begin 3) Set up the binomial factors like this: (x )(x ) 4) List the factor pairs of the LAST term *If the LAST term is POSITIVE, then the signs must be the same (both + or both -) *If the LAST term is NEGATIVE, then the signs must be different (one + and one -). 5) Find the pair whose sum is equal to the MIDDLE term 6) Check by multiplying the binomials (FOIL)

MAT 105 FALL 2008 Factor Completely

MAT 105 FALL 2008 Factor Completely

Factoring Trinomials in the Form MAT 105 FALL 2008 Factoring Trinomials in the Form The Trial & Check Method: Before you attempt to Un-FOIL 1) Always factor out the GCF first, if possible. 2) Write terms in descending order. Now we begin 3) Set up the binomial factors like this: ( x )( x ) 4) List the factor pairs of the FIRST term 5) List the factor pairs of the LAST term 6) Sub in possible factor pairs and ‘try’ them by multiplying the binomials (FOIL) until you find the winning combination; that is when O+I =MIDDLE term.

MAT 105 FALL 2008 Factor completely

MAT 105 FALL 2008 Factor completely

MAT 105 FALL 2008 Factor completely

A General Strategy for Factoring Polynomials MAT 105 FALL 2008 A General Strategy for Factoring Polynomials Before you begin to factor, make sure the terms are written in descending order of the exponents on one of the variables. Rearrange the terms, if necessary. Factor out all common factors (GCF). If your leading term is negative, factor out -1. If an expression has two terms, check for the difference of two squares: x2 - y2 = (x + y)(x - y) If an expression has three terms, attempt to factor it as a trinomial. If an expression has four terms, try factoring by grouping. Continue factoring until each individual factor is prime. You may need to use a factoring technique more than once. Check the results by multiplying the factors back out.

Section 6.5: Equivalent Fractions MAT 105 FALL 2008 Section 6.5: Equivalent Fractions The value of a fraction is unchanged if BOTH numerator and denominator are multiplied or divided by the same non-zero number. Equivalent fractions Equivalent fractions

An algebraic fraction is a ratio of two polynomials. MAT 105 FALL 2008 An algebraic fraction is a ratio of two polynomials. Some examples of algebraic fractions are: Algebraic fractions are also called rational expressions.

Simplifying Algebraic Fractions MAT 105 FALL 2008 Simplifying Algebraic Fractions A fraction is in its simplest form if the numerator and denominator have no common factors other than 1 or -1. (We say that the numerator and denominator are relatively prime.) We use terms like “reduce”, “simplify”, or “put into lowest terms”. Two simple steps for simplifying algebraic fractions: FACTOR the numerator and the denominator. Divide out (cancel) the common FACTORS of the numerator and the denominator.

Cancel only common factors. MAT 105 FALL 2008 WARNING: Cancel only common factors. DO NOT CANCEL TERMS! Example: NEVER EVER NEVER do this!!!!!!! Wrong! So very wrong!!

The correct way to simplify the rational expression MAT 105 FALL 2008 The correct way to simplify the rational expression Here is the plan: FACTOR the numerator and the denominator. Divide out any common FACTORS. Simplest form. Notice in this example because the value of the denominator would be 0. ,

Simplify the rational expression MAT 105 FALL 2008 Simplify the rational expression FACTOR the numerator and the denominator. Divide out any common FACTORS.

A Special Case The numerator and denominator are OPPOSITES. MAT 105 FALL 2008 A Special Case The numerator and denominator are OPPOSITES.

MAT 105 FALL 2008 Examples Simplify each fraction.

MAT 105 FALL 2008 Example Simplify each fraction.