Out line

1 Factoring Polynomials

• Terminology

• Factoring

2 Rational Expressions

• Definition

• Manipulating Rational Expressions

**Terms**

**Definition**

The terms of an algebraic expression are those elements that are

separated by addition (that is, by plus or minus signs).

The sign of the term is important!

**Monomials**

**Definition**

Terms that contain variables with only nonnegative integer

exponents are called monomials.

**Polynomials**

**Definition**

A polynomial is a monomial or a finite sum of monomials.

Examples of polynomials

**Factoring Out Monomials**

The simplest form of factoring polynomials is factoring
out the

highest common monomial factor.

• In x^{2} + 3x, each term contains a factor of x.

x^{2} + 3x = x(x + 3)

• In 3x^{4} + 12x^{2}, each term contains a factor of 3x^{2}.

3x^{4} + 12x^{2} = 3x^{2}(x^{2} + 4)

• In 12a^{2}b − 30ab + 18ab^{2}, each term contains a factor
of 6ab.

12a^{2}b − 30ab + 18ab^{2} = 6ab(2a − 5 + 3b)

**Factoring By Grouping**

Some times we can factor out a binomial by grouping terms in

pairs, and factoring a monomial out of each pair.

• x^{3} + 2x^{2} + 3x + 6

= (x^{3} + 2x^{2}) + (3x + 6)

= x^{2}(x + 2) + 3(x + 2)

= (x^{2} + 3)(x + 2)

• 3ac + 16b − 4a − 12bc

= 3ac − 12bc − 4a + 16b

= (3ac − 12bc) + (−4a + 16b)

= 3c(a − 4b) − 4(a − 4b)

= (3c − 4)(a − 4b)

**Factoring ax**^{2} + bx + c

Factoring Quadratics

To factor a quadratic of the form ax^{2} + bx + c:

Find a pair of numbers, say r and s, whose sum is b

(r + s = b), and whose product is ac (rs = ac).

Write the quadratic as ax^{2} + rx + sx + c.

Factor
by grouping.

Example: 4x^{2} + 11x + 6 a = 4, b = 11, c = 6

Find two numbers whose product is 24 and whose sum is 11.

r = 3, s = 8

Write the quadratic as 4x^{2} + 3x + 8x + 6.

Factor by grouping.

4x^{2} + 3x + 8x + 6

= (4x^{2} + 8x) + (3x + 6)

= 4x(x + 2) + 3(x + 2)

= (4x + 3)(x + 2)

**Special Factoring Patterns**

• x^{2} − y^{2} = (x + y)(x − y)

• x^{3} + y^{3} = (x + y)(x^{2} − xy + y^{2})

• x^{3} − y^{3} = (x − y)(x^{2} + xy + y^{2})

**Examples**

Factor 9x^{2} − 25 completely. (3x + 5)(3x − 5)

Factor 9x^{2}y^{2} − 64 completely. (3xy + 8)(3xy − 8)

Factor a^{2} + 5a − 24 completely. (a + 8)(a − 3)

Factor 2x^{2} − 7x − 30 completely. (2x + 5)(x − 6)

Factor x^{3} + 64 completely. (x + 4)(x^{2} − 4x + 16)

Factor 2n^{2} − n − 5 completely. Not factorable

**Definition**

**Definition**

The quotient of two polynomials is called a rational expression.

Examples

We will assume that all denominators represent nonzero
real

numbers (so we needn’t always write things like “x ≠−2” or

“x ≠1/3”).

**Simplifying**

• Factor anything you can.

• Cancel factors if possible.

• Remember that rational expressions are just fractions.

We really need to work examples.

**Examples**

Examples

Simplify

Simplify

Simplify

Simplify

5

Simplify