5-4 Factoring Polynomials Objectives: Students will be able to:
1)Factor polynomials 2)Simplify polynomial quotients by
factoring
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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.
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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.
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Example 1: Finding a GCF Example 1: Find the GCF of each set of
monomials. a)8, 12b) 10, 21c) 24, 60, 36 4112
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Ex 1: Finding GCFs 2x 2 4x 3x 2 6a 2 b 3xy 2
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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
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Ex 2: Factoring By Distributive Property Factor each
polynomial.
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Try these.
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Grouping Grouping is a factoring technique used when a
polynomial contains four or more terms.
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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)
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Example 3: Factor each polynomial.
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Ex 3: Factor each polynomial.
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Ex 3: Continued.
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Ex 3: Cont.
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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.]
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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.
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Ex4: Factoring Trinomials
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Example 4: Factor each polynomial
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Try some more
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Try these.
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More Examples
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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.
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Ex 5: Factor each polynomial
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Ex 5: GCF first!
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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.
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Factoring Differences of Squares
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GCF first!!
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Factoring Differences of Squares
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Sum/Difference of Cubes
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Try these
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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.
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Ex1: Simplify Factor Numerator and Denominator! Eliminate
Common Factors in Numerator and Denominator!
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Ex 2: Simplify
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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
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To recap: Always try and factor out a GCF first, if possible.
It will make life much easier.