5-4 factoring polynomials objectives: students will be able to: 1)factor polynomials 2)simplify...

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  • Slide 1
  • 5-4 Factoring Polynomials Objectives: Students will be able to: 1)Factor polynomials 2)Simplify polynomial quotients by factoring
  • Slide 2
  • Factoring There are various different techniques used to factor polynomials. The technique(s) used depend on the number of terms in the polynomial, and what those terms are. Throughout this section we will examine different factoring techniques and how to utilize one or more of those techniques to factor a polynomial.
  • Slide 3
  • What is a GCF Greatest common factor (GCF): largest factor that all terms have in common You can find the GCF for a polynomial of two or more terms.
  • Slide 4
  • Example 1: Finding a GCF Example 1: Find the GCF of each set of monomials. a)8, 12b) 10, 21c) 24, 60, 36 4112
  • Slide 5
  • Ex 1: Finding GCFs 2x 2 4x 3x 2 6a 2 b 3xy 2
  • Slide 6
  • Factoring a Polynomial w/GCF 1.Determine what the GCF of the terms is, and factor that out 2.Rewrite the expression using the distributive property
  • Slide 7
  • Ex 2: Factoring By Distributive Property Factor each polynomial.
  • Slide 8
  • Try these.
  • Slide 9
  • Grouping Grouping is a factoring technique used when a polynomial contains four or more terms.
  • Slide 10
  • Steps for Factoring By Grouping 1.Group terms with common factors (separate the polynomial expression into the sum of two separate expressions) 2.Factor the GCF out of each expression 3.Rewrite the expression using the distributive property (factor into a binomial multiplied by a binomial)
  • Slide 11
  • Example 3: Factor each polynomial.
  • Slide 12
  • Ex 3: Factor each polynomial.
  • Slide 13
  • Ex 3: Continued.
  • Slide 14
  • Ex 3: Cont.
  • Slide 15
  • Factoring Trinomials The standard form for a trinomial is: The goal of factoring a trinomial is to factor it into two binomials. [If we re-multiplied the binomials together, that should get us back to the original trinomial.]
  • Slide 16
  • Steps to factor a Trinomial Steps for factoring a trinomial 1)Multiply a * c 2) Look for factors of the product in step 1 that add to give you the b term. 3) Rewrite the b term using these two factors. 4) Factor by grouping.
  • Slide 17
  • Ex4: Factoring Trinomials
  • Slide 18
  • Example 4: Factor each polynomial
  • Slide 19
  • Try some more
  • Slide 20
  • Slide 21
  • Try these.
  • Slide 22
  • More Examples
  • Slide 23
  • Slide 24
  • Look For GCF first! There are instances when a polynomial will have a GCF that can be factored out first. Doing so will make factoring a trinomial much easier.
  • Slide 25
  • Ex 5: Factor each polynomial
  • Slide 26
  • Slide 27
  • Ex 5: GCF first!
  • Slide 28
  • Additional Factoring Techniques There are certain binomials that are factorable, but cannot be factored using any of the previous factoring techniques. These binomials deal with perfect square factors or perfect cube factors.
  • Slide 29
  • Factoring Differences of Squares
  • Slide 30
  • Slide 31
  • Slide 32
  • GCF first!!
  • Slide 33
  • Factoring Differences of Squares
  • Slide 34
  • Slide 35
  • Sum/Difference of Cubes
  • Slide 36
  • Slide 37
  • Try these
  • Slide 38
  • Slide 39
  • Slide 40
  • Simplifying Polynomial Quotients In the previous section (5-3), we learned to simplify the quotient of two polynomials using long division or synthetic division. Some quotients can be simplified using factoring. To do so: 1) factor the numerator (if possible) 2) factor the denominator (if possible) 3) reduce the fraction TIP: Be sure to check for values that the variable cannot equal. Remember that the denominator of a fraction can never be zero.
  • Slide 41
  • Ex1: Simplify Factor Numerator and Denominator! Eliminate Common Factors in Numerator and Denominator!
  • Slide 42
  • Ex 2: Simplify
  • Slide 43
  • Ex 3: Simplify In order to eliminate common factors, one must be in the numerator an the other in the denominator. This expression cannot be simplified further
  • Slide 44
  • To recap: Always try and factor out a GCF first, if possible. It will make life much easier.