R.5 Factoring Polynomials

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Chapter R Section 5 Factoring Polynomials

• In this section, we will
• Factor out the GCF (Greatest Common Factor)
• Factor by Grouping
• Factor Trinomials of the Form
• Factor Trinomials of the Form
• Factor the Sum and Difference of Two Perfect
  Cubes

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In the previous section, we learned how to
multiply polynomials in this section, we will
reverse the operation of multiplication by
finding the factors of a known product. If one
number a divides evenly into another number b,
then a is called a factor of b. example
Because 3 divides evenly into 24, 3is a factor of
24. multiply the polynomials
factor the polynomial
We are, in essence, un-distributing
R.5 Factoring Polynomials Factor Out the GCF
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Examples Factor out the GCF (greatest common
factor).
R.5 Factoring Polynomials Factor Out the GCF
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Examples Factor by grouping.
R.5 Factoring Polynomials Factor by Grouping
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• multiply the polynomials
  factor the polynomial
• Factoring Trinomials with a leading coefficient
  of 1
• Write the trinomial in descending order
• (for the first variable alphabetically)
• Write the factorizations of the third term of the
  trinomial
• Pick the factorization where the sum of the
  factors is equal to the coefficient of the middle
  term.

sum
product
We are, in essence, un-FOIL-ing
R.5 Factoring Polynomials Factor Trinomials of
the Form x bx c
2
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Examples Factor each trinomial.
R.5 Factoring Polynomials Factor Trinomials of
the Form x bx c
2
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Examples Factor each trinomial
R.5 Factoring Polynomials Factor Trinomials of
the Form x bx c
2
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Factoring Trinomials with a leading coefficient
other than 1 (2 methods)

• Key Number Method
• Write the trinomial in descending order
• Find the key number
• Find two factors of the key number whose sum is b
• Use those factors as the coefficients of two
  terms to replace the middle term
• Factor by grouping

• Guess-and-Check Method
• Write the trinomial in descending order
• Write the factorizations that you know
• Systematically guess-and-check factorization
  possibilities where the sum of the factors is
  equal to the coefficient of the middle term.
• (Be sure to check!)

R.5 Factoring Polynomials Factor Trinomials of
the Form ax bx c
2
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Examples Factor each trinomial.
If a trinomial has the form
with integer coefficients and . we
can test to see whether it is factorable If the
value of is a perfect square
(e.g. 0, 1, 4, 9, 16,) then the trinomial can be
factored using integers.
R.5 Factoring Polynomials Factor Trinomials of
the Form ax bx c
2
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Optional If you know that a trinomial is
factorable (because you used the discriminant
, but you cannot find the actual
factorsthe answer can be found by using the
graphing utility on your TI and using the
zeros.

• Example We will find the factors of the
  trinomial using the TI
• Enter the expression to be factored into
• Press
• (the viewing window may need to be adjusted in
  order to see where
  the graph crosses the x-axis)
• Press CALC menu
• For each zero, select 2zero and answer the three
  questions asked
• Algebra

Y
GRAPH
2ND
TRACE
Check to see if the factors are correct!
R.5 Factoring Polynomials Factor Trinomials of
the Form ax bx c
2
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sop
Examples Factor the sum or difference of two
cubes.
R.5 Factoring Polynomials Factor the Sum and
Difference of Two Perfect Cubes
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Factor-ific Flow Chart Cut off this slide and
use as a reference.
Write the polynomial in descending order (for the
first variable alphabetically)
Is the leading coefficient
negative?
yes
Factor out a -1 (or GCF)
Question 2
no
Is there a common
factor?
Question 2
yes
Factor out the GCF
Question 3
no
Determine if it a difference of squares,
difference of cubes or sum of cubes?
two
How many terms are there?
Question 3
three
Try factoring by trial-and-error (or use key
number method)
four
Try factoring by grouping.
Check each of the factors to see if we can factor
further.
R.5 Factoring Polynomials
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Examples Factor each polynomial completely.
R.5 Factoring Polynomials
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Examples Factor each polynomial completely.
R.5 Factoring Polynomials
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Independent Practice You learn math by doing
math. The best way to learn math is to practice,
practice, practice. The assigned homework
examples provide you with an opportunity to
practice. Be sure to complete every assigned
problem (or more if you need additional
practice). Check your answers to the
odd-numbered problems in the back of the text to
see whether you have correctly solved each
problem rework all problems that are
incorrect. Read pp. 49-55 Homework pp. 56-57
9-21 odds, 25, 27, 29, 33,
35, 37, 43-53 odds, 59-71
odds, 79-107 odds
Pretend youre starring in a reality show about a
kid who can make his dreams come true if he works
hard and gets good grades.
R.5 Factoring Polynomials
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R.5 Factoring Polynomials