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Chapter 3
Expressions and Polynomials
3.1 Evaluating Expressions

Differentiate between an expression and an equation
Equation: A mathematical relationship that contains an equal sign
Expression : A collection of constants, variables, and Operations
F = ma is an equation , whereas ma is an expression
5 + 6 = 11, is an equation, where as 5+6 is an
Expression
 To evaluate expression
1. Replace the variables with the corresponding given values
2. Calculate using the order of operations agreement
Division Properties
When zero is the divisor with any dividend other than zero, the quotient is
Undefined
n/0 = undefined when n = 0
0/n = 0, when n = 0
0/0 = indeterminate
3.2 Introduction to Polynomials
 Identify monomials
Monomial or Term : An algebraic expression that is a constant, or a
product of a constant and variables that are raised to whole –number
Powers.
x6 , x2y, 5x, -4xy3 , x3 y 3
 Identify the coefficient and degree of a monomial
Coefficient : The numerical factor in a monomial
Degree of a monomial : The sum of the exponents on all variables in a monomial
 Identify the like terms
Like terms: Monomials that have the same variables raised to the same exponents
x6 + 2 x6 = 3 x6
2x2y + 3 x2y = 5 x2y
- 4xy3 + 6xy3 = 2 xy3
x3 y 3 + 3x3 y 3
= 4 x3 y 3
x 3 y 3 - 5 x3 y 3
= -4 x3 y 3
Polynomials are algebraic expressions that are similar to
whole numbers written in expanded notation
For example
2394= 2.103+ 3.102+9.10+4 Expanded form , written with base 10
2x 3+ 3x2 + 9x + 4 Polynomial form , written with base x
 Identify the polynomials and their terms
Polynomial : A monomial or an expression that can be written as a sum of Monomials
Polynomial in one variable: A polynomial with only one variable
2x 3+ 3x2 + 9x + 4
Multivariable Polynomial: A polynomial with more than one variable
2x 3 y+ 3z2
 Identify the degree of a polynomial
Degree of a multiple-term polynomial: The greatest degree of all the terms that make
up the polynomial
2x 3+ 3x2 + 9x + 4
To write a polynomial in descending order of degree, place the term with greatest degree
first, then the term with the
next greatest degree, and so on.
2x 3+ 3x2 + 9x + 4
3.3 Simplifying the polynomials
To combine the like term
1.
Add or subtract the coefficients
2.
Keep the variables and their exponents the same
Examples
3x + 8x + 9x = 20x
2x2 + 5x2 = 7x2
- 8y3 + 6y3 = -2y3
3x2 – 5 + 4x 3 + 8x + 7x 2- 3x + 9 Combine like terms
4x 3 +7x 2 + 3x2 + 8x – 3x + 9 – 5
= 4x 3 + 10x 2
+ 5x
+4
3.4 Adding and Subtracting Polynomials
 Add polynomials in one variable
(2x 3+ 3x2 - x2 + 9x + 1) + (4x2 - 7x + 4 )
= 2x 3 + 3x2 + 4x2 - 9x – 7x + 1 + 4
= 2x 3 + 7x2 – 16x + 5
 Write an expression for the perimeter of a given shape
3x + 2
Perimeter
= 2( Length + width)
= 2( 2x + 1 + 3x + 2)
= 2( 2x + 3x + 1 + 2) ( combine like term)
= 2(5x + 3)
= 10x + 6
 Subtract polynomials in one variable
7x2 + 10x + 5 – (2x2 + 6x + 5 )
= 7x2 - 2x2 + 10x - 6x + 5 – 5
= 5x2 + 4x + 0
2x + 1
3.5 Multiplying Polynomials
Rule – When multiplying exponential forms that have he same base, we can add the
exponent and keep the same base
2 2 . 23 = 4. 8 = 32
2.2. 2.2.2
25 = 32
Alternative way 22 . 23 = 22+3 = 25
(x3 ) ( x4 )
=( x.x.x)(x.x.x.x)
=x7
x3 means 3x’s, x4 means 4x’s
Total 7x’s
Rules
Multiply monomials
1.
2.
3.
When multiplying exponential forms that have the same base, we can add the exponents
and keep the same base
n a nb = na +b
Simplify monomials raised to a power
To simplify an exponential form raised to a power, we can multiply the exponents and
keep the same base
( n a ) b = nab
Multiply polynomials
To multiply a polynomial by a monomial, use the distributive property to multiply each
term in the polynomial by the monomial
3x( 2x2 + 3x + 1)
6x3
9x2
3x
Multiply two polynomials:
4.
Multiply every term in the first polynomial by every term in the second polynomial
Combine like terms
(x + 6) (x + 7) = x2 + 6x + 7x + 42 = x2 + 13x + 42
5.
The product of two conjugates is a difference of two squares2x
(a + b)(a - b) = a2 - b2
3.6 Prime Numbers and GCF
Determine if a number is prime, composite, or neither
Prime number: A natural number other than 1 that has
exactly two different factors
List of primes : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47…….
Composite Number: A natural number that has factors other
than 1 and itself
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25
0 and 1 neither prime nor composite
To determine if a given number is prime or composite, divide the given
number by the primes on the list of prime numbers and consider the
results
Prime Factorization
To find the prime factorization of a number, use a factor
Tree
1.
2.
Draw two branches below the number
Place two factors that multiply to equal the given number at the end of the
two branches
Repeat steps 1 and 2 for every composite factor.
Place all the prime factors together in a multiplication sentence
3.
4.
84
2
84
42
2
21
3
7
4
2 2
21
3 7
84
7
84
12
2
3
6
2
3
28
2 14
2
7
Find all possible factors of a given number
Find the greatest common factor of a given set of numbers using prime
factorization.
24 = 2.2.2.3 = 23 .3
60= 2.2.3.5 = 22.3.5
GCF = 2.2.3 = 22 .3 = 12
Set of monomials
12x4 and 9x3
12x4 = 22 .3.x4
9x3 = 32 . X3
GCF = 3x3
3.7 Introduction to factoring
 Divide monomials
Rules
When dividing exponentials forms that have the same base, we can
subtract the divisor’s exponents from the dividend’s exponent and keep
the same base.
na/nb= n a – b , where n = b
Example x4 / x2 =x4-2=x2
4x4 / 2x3 = 2 x4-3= 2x
But 0a /0b= Indeterminate
Any base other than 0 power simplifies to the number 1
n0 = 1, when n = 0
00 = Indeterminate
Procedure
To divide monomials:
1.
Divide the coefficients.
3.
For like bases, subtract the exponent of the divisor base from the
exponent of the dividend base and keep the base. If the bases have
the same exponent, then they divide out, becoming 1.
4.
Bases in the dividend that have no like base in the divisor are
written unchanged in the quotient.

a)
b)
To divide a polynomial by a monomial.
Divide each term in the polynomial dividend by the monomial divisor
Simplify
Factor the GCF out of a polynomial
Procedure
To factor a monomial GCF out of a given polynomial
1. Find the GCF of the terms that make up the polynomial
2. Rewrite the polynomial as a product of the GCF and
parentheses that contain the quotient of the given
polynomial and the GCF.
Given polynomial = GCF
3.
Simplify in the parentheses
Given polynomial
GCF
Examples
Divide
1.
16x4 - 8x3 + 4x2
4x2
= 16x4
- 8x3
4x2
4x2
= 4x2
2.
-
2x
+ 4x2 (Divide each term by 4x2 )
4x2
+ 1
Factor
18y – 6
The GCF of 18 and 6 = 6
= 6(18y – 6)
6
= 6 ( 18y - 6) ( Divide each term of the polynomial by the GCF )
6
6
= 6(3y – 1)
( Simplify the paranthesis)
Check
6(3y – 1) ( Distributive property)
= 18y - 6