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5.4 Factor and Solve Polynomial Equations
Goal Factor and solve other polynomial equations.
Your Notes
VOCABULARY
Prime polynomial
A polynomial with two or more terms that cannot be written as a product of polynomials
of lesser degree using only integer coefficients and constants and the only common
factors of its terms are 1 and 1
Factored completely
A polynomial is factored completely if it is written as a monomial or the product of a
monomial and one or more prime polynomials.
Factor by grouping
A method used to factor some polynomials with pairs of terms that have a common
monomial factor
Quadratic form
An expression of the form au2 + bu + c, where u is any expression in x
FACTORING POLYNOMIALS
Definition A polynomial with two or more terms is a prime polynomial if it _cannot_ be
written as a product of polynomials of lesser degree using only integer coefficients and
constants and if the only common factors of its terms are _1_ and _1_.
Example 16x2  4x + 8 _is not_ a prime polynomial because _4_ is a common factor of
all its terms.
Definition A polynomial is factored completely if it is written as a monomial or the
product of a monomial and one or more _prime_ polynomials.
Example (x + 2)(x2  5x + 6) is not factored completely because
x2  5x + 6 = _(x  2) (x  3)_ .
Your Notes
SPECIAL FACTORING PATTERNS
Sum of Two Cubes
a3 + b3 = (a + b)(a2  ab + b2)
Example
x3 + 8 = (x + 2)(_x2  2x + 4_)
Difference of Two Cubes
a3  b3 = (a  b)(a2 + ab + b2)
Example
8x3  1 = (2x  1)(_4x2 + 2x + 1_)
Example 1
Factor the sum or difference of two cubes
Factor the polynomial completely.
a. z3  125 = z3  _53 _
Difference of
two cubes
= (z  _5_ )(_ z2 + 5z + 25_ )
b. 81y4 + 192y = 3y(_27y3 + 64_)
Factor common
monomial.
3
3
= 3y[_(3y) _ + _4 _]
Sum of two
cubes
2
= 3y(_3y + 4_)(_9y  12y + 16_)
Checkpoint Factor the polynomial completely.
1. 8x3 + 64
8(x + 2){x2  2x+ 4)
Example 2
Factor by grouping
Factor the polynomial x3  2x2  9x + 18 completely.
x3  2x2  9x + 18
Factor by grouping.
= x2(_x  2_)  9(_x  2_)
2
Distributive property
= _(x  9)(x  2)_
Difference of two
= _(x + 3)(x  3)(x  2)_
squares
Your Notes
Example 3
Factor polynomials in quadratic form
Factor completely: (a) 16x4  256 and (b) 3y7 - 15y5 + 18y3.
a. 16x4  256 = (_4x2_)2  _16 2_
= _(4x2 + 16)(4x2  16)_
= _(4x2 + 16)(2x + 4)(2x  4)_
7
5
b. 3y  15y + 18y3 = 3y3(_ y4  5y2 + 6 _)
= _3y3(y2  3)(y2  2)_
Checkpoint Factor each polynomial completely.
2.
x3 + 2x2  25x  50
(x + 5)(x  5)(x + 2)
3.
x4  14x2 + 45
(x2  5)(x + 3)(x  3)
Example 4
Solve a polynomial equation
What are the real-number solutions of the equation x4 + 9 = 10x2?
x4 + 9 = 10x2
_x4  10x2 + 9 = 0
_(x2  9)(x2  1) = 0
_(x + 3)(x  3)(x + 1)(x  1)_ = 0
x = _3_ , x = _3_, x = _1_ , x = _1_
The solutions are _3, 3, 1, and 1_ .
Write original
equation.
Write in standard
form.
Factor trinomial.
Difference of two
squares
Zero product
property
Checkpoint Find the real-number solutions.
4. 2x5 + 24x = 14x3
0, , 3 ,
3 , 2, 2
Homework
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