Download Factoring PowerPoint

FACTORING
Objective: To factor polynomials using a variety of
methods
Definitions:
 Product:
 The result of a multiplication problem
 Factor:
 The numbers and/or terms that are being multiplied
 Factoring:
 The process of breaking a number or polynomial into its
smallest factors
Types of Factoring
 Greatest Common Factor
 Difference of Squares
 Perfect Square Trinomial
 Factoring a trinomial into two binomials (using guess and
check)
 Sum/Difference of Cubes
Greatest Common Factor
 The GCF is the largest monomial that will divide evenly into
EVERY term of the polynomial
 You should always look for a GCF first when doing any factoring
problem
 Ex.
 In 20x4 + 35x2 the GCF is 5x2
 In 32a3b5 – 24a2b7 – 40ab8 the GCF is 8ab5
 When checking for a GCF, put the polynomial in standard form.
If the leading coefficient is negative, then the GCF should be
negative.
Greatest Common Factor
 Once you have determined the GCF, you will divide it out of
each term. The result should be written as
 GCF(polynomial result with GCF factored out)
 Example:
 20x4 + 35x2
GCF = 5x2
 20x 4  35x2
2

4x
7
2
5x

 20x4 + 35x2 = 5x2(4x2 +7)
Greatest Common Factor
 Example:
 6a3b5 – 9a4b6 + 15ab8 – 18a2b7
 Put in standard form:
 GCF:
-9a4b6 + 6a3b5 – 18a2b7 + 15ab8
-3ab5
 Factored form:
-3ab5(3a3b – 2a2 + 6ab2 – 5b3)
 Example:
 12m4 – 16m3 + 24m2
 GCF:
4m2
 Factored form:
4m2(3m2 – 4m + 6)
Factoring a Difference of Perfect Squares
 Multiply:
(2x – 3)(2x + 3)
 4x2 + 6x – 6x – 9
 4x2 – 9
 4x2 – 9 is known as a Difference of Perfect Squares
 A Difference of Perfect Squares is a BINOMIAL where both
terms are perfect squares and the terms are subtracted.
 A Difference of Perfect Squares a2 – b2 is factored into two
binomials:
 (a + b)(a – b)
Factoring a Difference of Perfect Squares
 Example:
 16m2 – 25
 (4m + 5)(4m – 5)
 Example:
 18a3 – 98a
 GCF:
2a
 Factor:
2a(9a2 – 49)  This is a difference of squares
 Final factored form:
2a(3a + 7)(3a – 7)
Factoring a Perfect Square Trinomial
 Multiply:
(2x + 3)2
 (2x + 3)(2x + 3)
 4x2 + 6x + 6x + 9
 4x2 + 12x + 9
 A Perfect Square Trinomial is a TRINOMIAL where the first and last
terms are perfect squares and the middle term is twice the product
of the square roots of the first and last terms (in the example above
the square root of the first term is 2x, the square root of the last term
is 3, and the middle term is equal to 2  2x  3 or 12x).
 The general form of a Perfect Square Trinomial is either a2 + 2ab + b2
or a2 – 2ab + b2 and factors to either (a + b)2 or (a – b)2
Factoring a Perfect Square Trinomial
 Factor:
x2 + 10x + 25
 The first and last terms are perfect squares, and the middle term is
double the product of the square roots of the first and last terms.
 Factored form:
 Factor:
(x + 5)2
9x2 – 24x + 16
 The first and last terms are perfect squares (3x and 4) and the middle
term is double their product (2  12x)
 Factored form:
 Factor:
(3x – 4)2
25x2 – 30x – 9
 The first and last terms are perfect squares, but the last term is negative
so this IS NOT a Perfect Square Trinomial and cannot be factored using
the rules for a Perfect Square Trinomial
Factoring a General Trinomial
 Multiply:
(x + 3)(x – 5)
 x2 – 5x + 3x – 15
 x2 – 2x – 15
 (x + 3) and (x – 5) are the factors of x2 - 2x – 15
 When factoring a trinomial, first check to see if there is a GCF or
if it is a Perfect Square Trinomial. If not, the easiest method is
to guess and check to find the factors.
Factoring a General Trinomial
 Example:
Factor x2 – 4x + 3
 Because the leading coefficient is 1, we can look for two numbers whose
product is 3 and whose sum is -4. Those two numbers are -3 and -1
 The two binomials whose product is x2 – 4x + 3 (factors) are (x – 3) and
(x – 1)
 The factored form is (x – 3)(x – 1)
 Example:
Factor x2 + 6x – 16
 Because the leading coefficient is 1, we can look for two numbers whose
product is -16 and whose sum is 6. Those two numbers are 8 and -2.
 The factored form is (x + 8)(x – 2)
Factoring a General Trinomial
 Example:
Factor 2x2 + 7x – 15
 Because the leading coefficient is not 1, this is a little more difficult to
factor.
 The first terms of each binomial factor must multiply to be 2x2, and the
second terms of each binomial factor must multiply to be -15. Guess
and check different combinations until you find one that works.
 Guess #1: (2x – 5)(x + 3)  This is not correct!
Check:
2x2 + 6x – 5x – 15 = 2x2 + x – 15
 Guess #2: (2x + 3)(x – 5)  This is not correct!
Check:
2x2 – 10x + 3x – 15 = 2x2 – 7x – 15
 Guess #3: (2x – 3)(x + 5)  YEA!!! We have found the correct factors!
Check:
2x2 + 10x – 3x – 15 = 2x2 + 7x - 15
Factoring a General Trinomial
 Example:
Factor 6x2 + 11x - 10
 The first terms in each binomial factor must multiply to be 6x2 and the last terms
in each binomial factor must multiply to be -10. Guess and check until you find
the combination that yields the correct middle term.
 (2x + 5)(3x – 2)
Check:
 Example:
6x2 – 4x + 15x – 10 = 6x2 + 11x – 10
Factor 10x3 + 35x2 + 15x
 There is a GCF of 5x
 5x(2x2 + 7x + 3)
 Guess and check until you find the right binomials to factor the trinomial
 2x2 + 7x + 3 factors to be (2x + 1)(x + 3)
 Final factored form: 5x(2x + 1)(x + 3)
Factoring a Sum or Difference of Cubes
 A Sum or Difference of Cubes is a BINOMIAL where each term is
a perfect cube.
 The Sum of Cubes a3 + b3 factors into a binomial multiplied by a
trinomial with the pattern:
 (a + b)(a2 – ab + b2)
 The Difference of Cubes a3 – b3 factors into a binomial
multiplied by a trinomial with the pattern:
 (a – b)(a2 + ab + b2)
Factoring a Sum or Difference of Cubes
 Example:
Factor 27x3 + 125
 This is a sum of cubes (3x and 5)
 The factors are (3x + 5)(9x2 – 15x + 25)
 Example:
Factor 16x4y – 54xy4
 There is a GCF of 2xy
 Factoring out the GCF yields 2xy(8x3 – 27y3)  This is a difference of cubes!
 The final factored form is 2xy(2x – 3y)(4x2 + 6xy + 9y2)