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Chapter 8
Polynomials and Factoring
Definitions
• Binomial- a polynomial of two terms
• Degree of polynomial- the greatest value of
exponent of any term of the polynomial
• Degree of monomial- the sum of the
exponents of the variables of a monomial
• Difference of two squares- a difference of two
squares is an expression of the form a²-b².
Factors to (a + b)(a - b)
Definitions
• Factoring by grouping- a method of factoring
that uses the distributive property to remove
a common binomial factor of two pairs of
terms
• Monomial- a real number, a variable, or a
product of a real number and one or more
variables with whole-number exponents
Definitons
• Perfect square trinomial- any trinomial of the
form a² + 2ab + b² or a² - 2ab + b²
• Polynomial- a monomial of the sum or difference
of two or more monomials. A quotient with a
variable in the denominator is not a polynomial
• Standard form of a polynomial- the form of a
polynomial that places the terms in descending
order by degree
• Trinomial- a polynomial of three terms
8.1
ADDING AND SUBTRACTING
POLYNOMIALS
Essential Understanding
• You can use monomials to form larger
expressions called polynomials. Polynomials
can be added or subtracted.
Degree of monomial
• The degree of a monomial is the sum of the
exponents of its variables only.
• The degree of a nonzero constant is 0.
– Zero means it has no degree
•
Finding the Degree
•
of Monomial
Any polynomial
•
whose largest term
•
has an exponent
•
that is not stated,
and no variables
•
has a degree of
•
zero
Examples:
11
⅞
12
½
∏
92
•
Finding the Degree
•
of Monomial
Any polynomial
•
whose largest term
•
has a variable with
•
an implicit
exponent of one
•
has a degree of
•
one
Examples:
x
z
a
y
b
p
•
Examples:
Finding the Degree
• Degree of 2
of Monomial
– 2xy
Any mononomial
whose largest term
– 4z²
has an exponent or
– 7pq
has multiple terms
with exponents has • Degree of 3
– xyz
the sum of the
exponents as its
– x³
degree
– Xy²
• Degree of 4
– x²y²
– wxyz
Combining Like Monomials
Example
Explanation
• To find degree, sum exponents
• Even if bases are not the same
• x²x²
– 2+2=4
• 6x³y²
– 3+2=5
• -7x⁴z²
– 4+2=6
• uvwxyz
– 1+1+1+1+1+1=6
• Combine exponents
– Term has a degree of four
• Combine exponents
– Term has a degree of five
• Combine exponents
– Term has a degree of six
• Combine exponents
– Term has a degree of six
Standard Form of Polynomial
Examples in Standard Form
• In standard form, the
exponents (degree) descend
as the terms are listed.
• 5x² + 7x - 10
• 12x³ - 6x + 9
• 19x⁴ - 8x³ - 5x
• x⁴ + x³ + x² + x + 1
Examples NOT in Standard Form
• An expression is not in
standard form if the
degrees do not descend in
order
• 6x-12x²
• 1-3x⁴+2x+x³
• 12 + 12x² + 12x⁴ + 12x
Degree of a Polynomial
• The degree of a polynomial in one variable is
the same as the degree of the monomial with
the greatest exponent.
• The degree of 3x⁴ + 5x² - 7x + 1 is four
Degree of Polynomial
Polynomial Degree
Name of
Degree
Number of
Terms
Name using
Number of
Terms
10
None (0)
Constant
One
monomial
9237238
None (0)
Constant
One
monomial
x
One (1)
Linear
One
monomial
7x-5
One (1)
Linear
Two
binomial
5x²
Two (2)
Quadratic
One
monomial
12x²+144
Two (2)
Quadratic
Two
binomial
3x²+9x+12
Two (2)
Quadratic
Three
trinomial
8x³
Three (3)
Cubic
One
monomial
4x³+10x
Three (3)
Cubic
Two
binomial
x³+3x²+x+1
Three (3)
Cubic
Four
polynomial
x⁴+x³+x²+x+1
Four (4)
Quartic
Five
polynomial
Classifying Polynomials
• To classify a polynomial
1.
2.
3.
4.
5.
6.
Ensure all like terms are combined
Find the term with the greatest exponent
List term with greatest exponent
Find term with next greatest exponent
List term with next greatest exponent
Continue process until you reach the constant
term or, if no constant term exists, the term
with the least greatest exponent
8.2
MULTIPLYING AND FACTORING
Essential Understanding
• You can use the distributive property to
multiply a monomial by a polynomial
Multiplying Polynomials
Greatest Common Factor
8.3
MULTIPLYING BINOMIALS
Essential Understanding
• There are several ways to find the product of
two binomials including models, algebra, and
tables.
Multiplying Binomials
Multiplying Binomials Using the
Distributive Property
Multiplying Binomials Using a Table
FOIL
• FOIL stands for first outer inner last
• This method does not work for multiplying
two polynomials with more than two terms
for each
8.4
MULTIPLYING SPECIAL CASES
Essential Understanding
• There are special rules you can use to simplify
the square of a binomial or the product of a
sum and difference.
• Squares of binomials have two forms:
– (a + b)²
– (a – b)²
The Square of a Binomial
• The square of a binomial is the square o the
first term plus twice the product of the two
terms plus the square of the last term.
• (a + b)² = a² + 2ab + b²
• (a – b)² = a² -2ab + b²
The Product of a sum and difference
• The product of the sum and difference of the
same two terms is the difference o their
squares.
• (a + b)(a – b) = a² - b²
8.5
FACTORING X²+BX+C
Essential Understanding
• You can write some trinomials of the form
x² + bx + c as the product of two binomials
Factoring x² + bx + c
• Use a table to list the pairs of factors of the
constant term c and the sums of those pairs of
factors.
8.6
FACTORING AX²+BX+C
Essential Understanding
• You can write some trinomials of the form
ax² + bx + c as the product of two binomials
Factoring When a∙c is Positive
Factoring When a∙c is Negative
8.7
FACTORING SPECIAL CASES
Essential Understanding
• You can factor some trinomials by “reversing”
the rules for multiplying special case binomials
that you learned in lesson 8.4
Factoring Perfect Square Trinomials
• Any trinomial of the form a² + 2ab + b² is a
perfect square trinomial because it is the
result of squaring a binomial.
• For example let a,b be real numbers:
– (a + b)² = (a + b)(a + b) = a² + 2ab + b²
– (a - b)² = (a - b)(a - b) = a² - 2ab + b²
Factoring a Difference of Two Squares
• For all real numbers a,b:
– a² - b² = (a + b)(a – b)
8.8
FACTORING BY GROUPING
Essential Understanding
• Some polynomials of a degree greater than 2
can be factored
Factoring Polynomials
1. Factor out the greatest common factor (GCF)
2. If the polynomial has two terms or three
terms, look for a difference of two squares, a
perfect square trinomial or a pair of binomial
factors
3. If the polynomial has four or more terms,
group terms and factor to find common
binomial factors.
4. As a final check, make sure there are no
common factors other than 1