FACTORING COMPLETELY A factorable polynomial with integer coefficients is factored completely when it is written as a product of unfactorable polynomials with integer coefficients Finding a common monomial factor ID: 917689
Download Presentation The PPT/PDF document "Factoring Polynomials Tanya Magana" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Factoring Polynomials
Tanya Magana
Slide2FACTORING COMPLETELY
A factorable polynomial with integer coefficients is factored completely when it is written as a product of
unfactorable
polynomials with integer coefficients.
Slide3Finding a common monomial factor
Slide4EXAMPLES
x^3 – 4x^2 – 5x
X^3 – 4x^2 – 5x = x(x^2-4x – 5)
Factor common monomial
= x(x-5) (x+1)
Factor trinomial
2. 3y^5 – 48y^3
3y^5 – 48y^3 = 3y^3(y^2 – 16)
Factor common monomial
= 3y^3(y – 4) (y + 4)
Difference of Two Squares Pattern
3. 5z^4 + 30z^3 + 45z^2
5z^4 + 30z^3 + 45z^2 = 5z^2(z^2 + 6z + 9)
Factor common monomial
= 5z^2(z+3)^2
Perfect Square Trinomial pattern
Slide5SPECIAL FACTORING PATTERN
SUM OF TWO CUBES
a^3 + b^3 = (a + b) (a^2 – ab + b^2)
DIFFERENCE OF TWO CUBES
a^3 – b^3 = (a - b)(a^2 + ab + b^2)
Slide6Factoring the sum or difference of two cubes
Slide7EXAMPLES
X^3 – 125
X^3 – 125 = x^3 – 5^3
Write as a^3 – b^3
= (x – 5) (x^2 + 5x + 25) Difference of Two Cubes Pattern
2. 16x^5 + 54x^2
16x^5 + 54x^2 = 2x^2 (8x^3 + 27)
Factor common monomial
= 2x^2 ((2x)^3 + 3^3)
Write 8x^3 + 27 as a^3 + b^3
= 2x^2(2x + 3) (4x^2 – 6x + 9)
Sum of two cubes Pattern
Slide8Factoring by grouping
Slide9Factor by grouping pairs of terms that have a common monomial factor.
Pattern is
ra
+
rb
+
sa
+
sb
=
r
(
a
+
b
) +
s
(
a
+
b
)
= (
r
+
s
) (
a
+
b
)
Example:
Z^3+5z^2 – 4z – 20 = z^2(z + 5) - 4(z + 5)
Factor by grouping
= (z^2 – 4) (z + 5)
Distributive Property
= (z – 2) (z + 2) (z + 5)
Difference of Two Squares
Slide10Factoring polynomials in quadratic form
Slide11EXAMPLES
3p^8 + 15p^5 + 18p^2
3p^8 + 15p^5 + 18p^2 = 3p^2(p^6 + 5p^3 + 6)
Factor common monomial
= 3p^2(p^3 + 3) (p^3 +
Factor trinomial in quadratic form
Slide12The factor theorem
Slide13A polynomial
f
(x) has a factor x – k if and only if
f
(k) = 0
F(x) = (x – k) * q(x) , x – k is a factor of f(x). This result is summarized by the factor theorem, a special case of the remainder theorem.
Slide14STRUCTURE
Slide15a^2 + 2ab + b^2 = ( a + b )^2
a^2 – 2ab + b^2 = (a – b)^2
EXAMPLES:
Show that the expression (x + y) (x - y) - 6x + 9 may be written as the difference of two squares and factor the expression
Factor the following completely: 2(x-2)^2 – 5(x-2) - 12
Factor the following completely: 10(2a-1)^2 – 19(2a-1) - 15
Slide16PRACTICE
Slide17X^3 – 2x^2 – 24x
2q^4 + 9q^3 – 18q^2
16t^7 + 250t^4
135z^11 – 1080z^8
X^3 – 8x^2 – 4x + 32
16n^3 + 32n^2 – n – 2
49k^4 – 9
4n^12 – 32n^7 + 48n^2
DETERMINE WHETER THE BINOMIAL IS A FACTOR OF THE POLYNOMIAL
F(x) = 2x^3 + 5x^2 – 37x – 60; x – 4
H(x) = 6x^5 – 15x^4 – 9x^3 ; x + 3