# Algebraic Expressions

**1.3 Algebraic Expressions
**

A

**polynomial**is an expression of the form:

The numbers are called

**coefficients .**

Each of the separate parts, such as or , is called a

**term**of the polynomial.

If there is only one term, it is called a monomial. For two terms , it is called a binomial . For three terms, a

trinomial .

The

**degree**of the polynomial is n: the highest power of x.

To add ( or subtract ) polynomials, we combine like terms (those that have the same variables raised to the

same powers).

Example: Find the difference (x

^{3}- 4x

^{2}+ 6x + 12) - (2x

^{3}+ x

^{2}- 4x - 6)

To multiply polynomials, we use the distributive laws . In particular, to multiply two binomials, we use the

FOIL method.

Example: Multiply (x - 7)(2x + 3)(x - 1)

**Special Product and Factoring Formulas**

1. Difference of Squares

(A + B)(A - B) = A

^{2}- B

^{2}

2. Perfect Squares

(a) (A + B)

^{2}= A

^{2}+ 2AB + B

^{2}

(b) (A - B)

^{2}= A

^{2}- 2AB + B

^{2}

3. Cubing a Sum or Difference

(a) (A + B)

^{3}= A

^{3}+ 3A

^{2}B + 3AB

^{2}+ B

^{3}

(b) (A - B)

^{3}= A

^{3}- 3A

^{2}B + 3AB

^{2}- B

^{3}

4. Sum or Difference of Cubes

(a) A

^{3}+ B

^{3}= (A + B)(A

^{2}- AB + B

^{2})

(b) A

^{3}- B

^{3}= (A - B)(A

^{2}+ AB + B

^{2})

**Examples: **Expand the following.

•

•
(x^{3} - 4)^{2}

•
(x^{2} - 2)^{3}

** Factoring Steps
**

1. Factor out all common factors.

2. See if you can use a special factoring formula .

3. See if you can factor by grouping.

4. Use trial and error.

**Examples:**Factor the following.

• 8x

^{2}- 24x + 18

• 36x

^{2}- 25

• x

^{3}- 27

• 2x^{2} + 11x - 21

•
x^{3} + 3x^{2} - x - 3

• (a^{2} + 2a)^{2} - 2(a^{2} + 2a) - 3

To factor expressions with rational exponents , first factor
out, if possible, the smallest power of x.

• Factor

Its easy to check if you factored correctly. Just multiply back out to double
check.

**1.4 Rational Expressions
**

A

**rational expression**is a fractional expression where both the numerator and denominator are polyno-

mials.

The

**domain**of any algebraic expression is the set of values that the variable can be.

So far, we have two things to look for to determine the domain:

• The denominator can 't be zero. If a value of x makes the denominator zero, we must exclude it from

the domain.

• For even powered roots (square roots, fourth roots, etc), whatever is under the radical must be ≥ 0.

**Examples:**Find the domains of the following rational expressions.

Working with rational expressions is just like working
with fractions.

To multiply rational expressions, factor the numerator and denominator ,
multiply, and then simplify by

cancelling common factors in the numerator and denominator.

To divide rational expressions, multiply by the reciprocal .

To add or subtract rational expressions, you MUST have a COMMON DENOMINATOR
(just like with

fractions).

**Examples**

** Compound fractions ** are fractional expressions where the
numerator and/or denominator are themselves

fractional expressions.

**Examples:**

**Rationalizing the Denominator or Numerator
**

As before, rationalizing means to get rid of any radicals. We do this by multiplying by the

**conjugate**. The

conjugate is usually found by just changing the sign of the second term.

For example, the conjugate of is . What's the point? When you multiply these two

expressions together you get an expression that has NO radicals:

**Examples**

• What are the conjugates of:

• Rationalize the denominator of
.

•
Rationalize the numerator of .

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