5.4 Factor and Solve Polynomial Equations

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)_ .

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. z^3 - 125 = z^3 - _5^3 _	Difference of two cubes
= (z - _5_ )(_ z^2 + 5z + 25_ )
b. 81y^4 + 192y = 3y(_27y^3 + 64_)	Factor common monomial.
= 3y[_(3y)^3_ + _4^3_]	Sum of two cubes
= 3y(_3y + 4_)(_9y^2 - 12y + 16_)

Example 2

Factor by grouping

Factor the polynomial x³ - 2x² - 9x + 18 completely.

x³ - 2x² - 9x + 18

= x^2(_x - 2_) - 9(_x - 2_)	Factor by grouping.
= _(x^2 - 9)(x - 2)_	Distributive property
= _(x + 3)(x - 3)(x - 2)_	Difference of two squares

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)_

b. 3y⁷ - 15y⁵ + 18y³ = 3y³(_ y⁴ - 5y² + 6 _)

= _3y³(y² - 3)(y² - 2)_

Example 4

Solve a polynomial equation

What are the real-number solutions of the equation x⁴ + 9 = 10x²?

x^4 + 9 = 10x^2	Write original equation.
_x^4 - 10x^2 + 9 = 0	Write in standard form.
_(x^2 - 9)(x^2 - 1) = 0	Factor trinomial.
_(x + 3)(x - 3)(x + 1)(x - 1)_ = 0	Difference of two squares
x = _-3_ , x = _3_, x = _-1_ , x = _1_	Zero product property
The solutions are _-3, 3, -1, and 1_ .