Presentation on theme: "MAT 105 FALL 2008 Review of Factoring and Algebraic Fractions"— Presentation transcript:
1MAT 105 FALL 2008 Review of Factoring and Algebraic Fractions Chapter 6Review of Factoringand Algebraic Fractions
2Section 6.2: Factoring: Common Factors and Difference of Squares MAT 105 FALL 2008Section 6.2: Factoring: Common Factors and Difference of SquaresFactoring is the reverse of multiplying.A polynomial or a factor is called _________________ if it contains no factors other than 1 or -1.
3THE FIRST STEP: Factoring Out the Greatest Common Monomial Factor MAT 105 FALL 2008THE FIRST STEP: Factoring Out the Greatest Common Monomial Factor
4Factoring the Difference of Perfect Squares MAT 105 FALL 2008Factoring the Difference of Perfect SquaresRecall:Difference of Squares:
5Factoring the Difference of Perfect Squares MAT 105 FALL 2008Factoring the Difference of Perfect Squares
6Factor Completely: HINT: Always check for a GCF first!! MAT 105 FALL 2008Factor Completely:HINT: Always check for a GCF first!!
7MAT 105 FALL 2008Factoring 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.
10I. Factoring Trinomials in the Form MAT 105 FALL 2008Section 6.3: Factoring TrinomialsI. Factoring Trinomials in the FormRecall:FLO + ITo factor a trinomial is to reverse the multiplication process (UnFOIL)
11Before you attempt to Un-FOIL MAT 105 FALL 2008Before you attempt to Un-FOIL1) Always factor out the GCF first, if possible.2) Write terms in descending order.Now we begin3) 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 term6) Check by multiplying the binomials (FOIL)
14Factoring Trinomials in the Form MAT 105 FALL 2008Factoring Trinomials in the FormThe Trial & Check Method:Before you attempt to Un-FOIL1) Always factor out the GCF first, if possible.2) Write terms in descending order.Now we begin3) Set up the binomial factors like this: ( x )( x )4) List the factor pairs of the FIRST term5) List the factor pairs of the LAST term6) 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.
18A General Strategy for Factoring Polynomials MAT 105 FALL 2008A General Strategy for Factoring PolynomialsBefore you begin to factor, make sure the terms are written in descendingorder 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.
19Section 6.5: Equivalent Fractions MAT 105 FALL 2008Section 6.5: Equivalent FractionsThe value of a fraction is unchanged if BOTH numerator and denominator are multiplied or divided by the same non-zero number.Equivalent fractionsEquivalent fractions
20An algebraic fraction is a ratio of two polynomials. MAT 105 FALL 2008An algebraic fraction is a ratio of two polynomials.Some examples of algebraic fractions are:Algebraic fractions are also called rational expressions.
21Simplifying Algebraic Fractions MAT 105 FALL 2008Simplifying Algebraic FractionsA 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.
22Cancel only common factors. MAT 105 FALL 2008WARNING:Cancel only common factors.DO NOT CANCEL TERMS!Example: NEVER EVER NEVER do this!!!!!!!Wrong! So very wrong!!
23The correct way to simplify the rational expression MAT 105 FALL 2008The correct way to simplify the rational expressionHere 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.,
24Simplify the rational expression MAT 105 FALL 2008Simplify the rational expressionFACTOR the numerator and the denominator.Divide out any common FACTORS.
25A Special Case The numerator and denominator are OPPOSITES. MAT 105 FALL 2008A Special CaseThe numerator and denominator are OPPOSITES.
26MAT 105 FALL 2008ExamplesSimplify each fraction.