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CHAPTER 3
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Integer Exponents

If a is a real number and n is a positive integer,
then an represents a as a factor n times in a product
 an =
a
a•a•a•…•a•a
is called the base
 n is called the exponent
Properties of Exponents

For any real number a, and integers m and n,

am x an = am+n

am ÷ an = am-n
Examples
Zero and Negative Exponents


For real numbers a and b, neither can be zero, and
integer n,
a0 = 1
1
=
𝑎
1
-n
 a = 𝑛
𝑎
 a-1

(𝑎𝑏)-n = (𝑎𝑏)n
Examples
Absolute Value

Absolute value measures the distance between the
number inside the absolute value, and zero

Always a positive answer

If number inside is positive, absolute value does nothing

If number inside is negative, absolute vale makes it
positive
Examples
Rational Exponents

If a is a real number, variable, or algebraic
expression and n is a positive integer n≥2, then
1
 an
=
𝑛
𝑎
 Provided

that
𝑛
𝑎 exists
If a is a real number and if m and n are integers
containing no common factor with n≥2, then
𝑚
a𝑛
=
𝑛
𝑎𝑚 = ( 𝑛 𝑎)m
 Provided
that
𝑛
𝑎 exists
Examples
Multiplication on Monomials and Binomials





Monomials – Polynomials with only one term
Binomials – Polynomials with two terms
If multiplying monomials together, multiply like terms
together
If multiplying a monomial with a binomial, multiply
the monomial by each term in the binomial
If multiplying binomials together, use FOIL method
Examples
Special Binomial Products

(x+a)(x-a) = x2 – a2

(x+a)2 = x2 + 2ax + a2
Perfect Square Trinomial

(x-a)2 = x2 – 2ax + a2
Perfect Square Trinomial
Difference of Two Squares
Examples
Factoring

When factoring always factor out the Greatest
Common Factor (GCF), if one exists
Factoring

After GCF, use knowledge of special binomial
products to factor
Factoring

If there are 4 terms, factor by grouping
Factoring

If a trinomial is being factored, follow the following steps to factor
Steps
Example
To factor a quadratic trinomial in the variable x:
Factor 5x – 6 + 6x2
1.Arrange the trinomial with the powers of x in
descending order
6x2 +5x - 6
2.Form the product of the second-degree term
and the constant term (first and third terms)
6x2(-6) = -36x2
3.Determine if there are 2 factors of the
product in step above that will sum to the
middle term of quadratic (if there are no such
factors, trinomial cannot be factored)
-36x2 = (-4x)(9x) and -4x + 9x = 5x
4.Replace the middle term from step 1 with the
sum of the two factors from step 3
6x2 +5x – 6 = 6x2 -4x + 9x - 6
5. Factor the four term polynomial by grouping
6x2 -4x + 9x – 6 = (6x2 -4x) + (9x – 6) =
2x(3x-2) + 3(3x-2) = (3x-2)(2x+3)
Example
Complex Numbers

The number a + bi, in which a and b are real
numbers, is said to be a complex number in
standard form. The a is the real part of the number,
and the bi is the imaginary part. If b=0, the number
a + bi = a is a real number, and if b≠0, the
number a + bi is an imaginary number. If a=0,
a + bi = bi, which is a pure imaginary number
i
= −1
Examples
Identify Complex Numbers




A + Bi
Real = A
Imaginary = A + Bi
Pure Imaginary = Bi
Homework

Page 172

1-31