Polynomials and
Factoring
CHAPTER 9
• This chapter presents a number of skills
necessary prerequisites to solving
equations.
• These skills involve combining,
simplifying, and factoring polynomials.
• Factoring will be made easier if students can
recognize patterns that frequently occur in
products.
Introduction
• Monomial: An expression that is a number, a variable, or
a product of a number and one or more variables.
• Examples: 12, y, -5x2y,
𝑐
3
• Degree of a monomial: The sum of the exponents of its
variables.
• For a non-zero constant, the degree is zero.
• Zero, itself, has no degree.
Adding and Subtracting
Polynomials (9.1)
Sample Problem
Find the degree of each monomial.
a)
2
x
3
b) 7x2y3
c) -4
Adding and Subtracting
Polynomials (9.1)
• Polynomial: A monomial or the sum or difference of two
or more monomials.
• Example: 3x4 + 5x2 – 7x +1
• This polynomial is written in standard form.
• Standard form of a polynomial: This is the form of a
polynomial when the degrees of its monomial terms are
written so that they decrease from left to tight.
• Degree of a polynomial: This will be the same as the
monomial with the greatest exponent.
Adding and Subtracting
Polynomials (9.1)
• After simplifying a polynomial by combining like terms,
we can name the polynomial based on its degree or the
number of monomials it contains.
Polynomial
Degree
Name Using
Degree
Number of
Terms
Name Using Number of
Terms
7x+4
1
Linear
2
Binomial
3x2+2x+1
2
Quadratic
3
Trinomial
4x3
3
Cubic
1
Monomial
9x4+11x
4
Fourth degree
2
Binomial
5
0
Constant
1
Monomial
Adding and Subtracting
Polynomials (9.1)
Sample Problem
Write each polynomial in standard form. Then name each
polynomial based on its degree and the number of its terms.
a) 5 – 2x
b) 3x4 – 4 + 2x2 + 5x4
Adding and Subtracting
Polynomials (9.1)
• Adding and subtracting polynomials simply require that
we add or subtract the like terms.
• Group the like terms then add or subtract.
• Remember, subtraction means to add the opposite, which
means we change the signs of each term then add
(Chapter1).
Sample Problem
Simplify the following polynomials
a) (4x2 + 6x + 7) + (2x2 – 9x +1)
b) (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11)
Adding and Subtracting
Polynomials (9.1)
• Multiplying polynomials require that we remember and
use the Distributive Property (Chapter 1).
• Example: 2x(3x + 1) = (2x)(3x) + (2x)(1) = 6x2 + 2x
• We can also use the Distributive Property for multiplying
powers with the same base when multiplying by a
monomial.
Sample Problem
Simplify
−4𝑦2(5𝑦4 − 3𝑦2 + 2)
Multiplying and Factoring
(9.2)
• Factoring requires us to reverse the multiplication
process.
• To factor a monomial from a polynomial, we have to first
find the greatest common factor (GCF) of its terms.
• The greatest common factor is the greatest factor that
divides evenly into each term of an expression.
• To factor a polynomial completely, you must factor until
there are no more common factors other than one.
Multiplying and Factoring
(9.2)
Sample Problem
a) Find the GCF of the terms of 4𝑥3 + 12𝑥2 − 8𝑥)
b) Factor: 3x3 – 12x2 + 15x
Multiplying and Factoring
(9.2)
• We can also use the Distributive Property to multiply two
binomials together.
Sample Problem
Simplify (2x + 3) (x + 4)
Multiplying Binomials
(9.3)
• One way to organize multiplying two binomials is to use
the acronym FOIL.
•
•
•
•
F: First
Sample Problem
O: Outside Simplify (3x – 5)(2x + 7)
I: Inside
L: Last
Multiplying Binomials
(9.3)
• FOIL is useful for multiplying two binomials, but it does
not work when multiplying a trinomial and a binomial.
• In these cases, we can use the either the horizontal method
or the vertical method of distribution to help us organize our
work.
Sample Problem
Simplify the product (4x2 + x – 6)(2x – 3)
Multiplying Binomials
(9.3)
The Square of a Binomial
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
• To square a binomial you can either use the above rule or
use the FOIL method.
• Let’s see hoe the FOIL method will give us the above
expressions.
Multiplying Special Cases
(9.4)
Sample Problem
Find each square.
a) (x + 7)2
b) (4k – 3)2
Multiplying Special Cases
(9.4)
The Difference of Squares
(a + b) (a – b) = a2 – b2
• Let’s use FOIL to derive the above rule.
Multiplying Special Cases
(9.4)
Sample Problem
Find the product of (t3 – 6)(t3 + 6)
Multiplying Special Cases
(9.4)
• To factor a trinomial of the form x2 + bx + c, you
must:
• Find two numbers that have the sum of b,
• Find two numbers that have the product of c.
• At first this process will be trial and error, but as
you do more problems patterns will arise that you
will recognize that will speed up the factoring
process.
• The end result from factoring will be two
binomials.
Factoring Trinomials of
2
the Type x + bx +c (9.5)
Sample Problem:
Factor x2 + 7x + 12
Factoring Trinomials of
2
the Type x + bx +c (9.5)
• To factor trinomials of the form x2 – bx + c:
• Following the same method as listed above, however,
inspect the negative factors of c to make sure that the sum
will be a negative number.
Sample Problem
Factor d2 – 17d + 42
Factoring Trinomials of
2
the Type x + bx +c (9.5)
• To factor trinomials of the form x2 + bx – c or x2 – bx – c
• Following the same method as listed above, however,
inspect the negative factors that are both positive and
negative
Sample Problem
Factor:
a) m2 + 6m – 27
b) p2 – 3p – 18
Factoring Trinomials of
2
the Type x + bx +c (9.5)
• Factoring trinomials that have more than one variable is
also possible.
• The first term will be the square of the first variable.
• The middle term includes both variables.
• The last term includes the square of the second variable.
Sample Problem
Factor:
a) h2 – 4hk – 77k2
Factoring Trinomials of
2
the Type x + bx +c (9.5)
• To factor trinomials of the form ax2 + bx +c involves a
little more work, but is not much more difficult.
• To find the binomials that are the factor of the above
trinomial:
• Look for two numbers that will give the product a.
• Look for two numbers that will give the product c.
• The sum of the two products will give b.
Factoring Trinomials of
2
the Type ax + bx +c (9.6)
Sample Problem
Factor 6n2 + 23n + 7
Factoring Trinomials of
2
the Type ax + bx +c (9.6)
Sample Problem
Factor 7x2 + 26x – 8
Factoring Trinomials of
2
the Type ax + bx +c (9.6)
• Some polynomials require more than one factoring step.
• In this case, continue factoring until there are no common
factors other than 1.
• If a trinomial has a common monomial factor, factor it out
before trying to find the binomial factors.
Sample Problem
Factor completely:
a) 20x2 + 80x + 35
Factoring Trinomials of
2
the Type ax + bx +c (9.6)
• Perfect square trinomial: Any trinomial that takes the form a2
+ 2ab + b2 or a2 – 2ab + b2.
• Perfect square trinomials factor to give identical binomial
factors.
• Factoring perfect square trinomials is made a lot easier if you
learn to recognize when a perfect square trinomial appears.
Perfect-Square Trinomials
a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2
a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2
Factoring Special Cases
(9.7)
• To recognize a perfect square trinomial:
• The first and the last terms can both be written as the
product of two identical factors.
• The middle term is twice the product of one factor from the
first term and one factor from the last term.
• Example:
•
•
•
•
4x2 + 12x + 9
First term: 2x • 2x
Last term: 3 • 3
Middle term: 2(2x • 3)
Factoring Special Cases
(9.7)
Sample Problem
Factor each expression:
a) x2 – 8x + 16
b) 9g2 + 12g + 4
Factoring Special Cases
(9.7)
• Factoring can also be made easier when you recognize
that you have a difference of two squares:
Difference of Two Squares
a2 – b2 = (a + b)(a – b)
Sample Problem
Factor: x2 – 64
Factoring Special Cases
(9.7)
• Sometimes, to recognize that we have a difference of
square, we will have to factor out the greatest common
factor first.
Sample Problem
Factor: 10x2 – 40
Factoring Special Cases
(9.7)
• We can use the Distributive Property to factor by
grouping if two groups of terms have the same factor.
• To factor by grouping:
• Group terms and use the Distributive Property to see if a
common term exits;
• Look for a common binomial factor of two pairs of terms;
• Factor out the common binomial.
• This method works for larger polynomials (i.e. four term
polynomials).
Factoring by Grouping
(9.8)
Sample Problem
Factor 4n3 + 8n2 – 5n – 10
Factoring by Grouping
(9.8)
• Sometimes before factoring by grouping, you may need
to factor the GCF of all terms of a polynomial.
Sample Problem
Factor 12p4 + 10p3 – 36p2 – 30p.
Factoring by Grouping
(9.8)
• We can also use factor by grouping to factor trinomials of
the form ax2 + bx + c.
• We can use this method of factoring in situations in which
we cannot quickly factor a trinomial using the methods
discussed in Lesson 9.6.
• To factor a trinomial by grouping:
•
•
•
•
Find the product ac;
Find the two factors of ac that have the sum b;
Rewrite the trinomial using this sum;
Factor by grouping.
Factoring by Grouping
(9.8)
Sample Problem
Factor 24q2 + 25q – 25 .
Factoring by Grouping
(9.8)
Summary of Factoring Polynomials:
1. Factor out the GCF.
2. If the polynomial has two terms or three terms, look for
a difference of two squares, a product of two squares, or
a pair of binomial factors.
3. If there are 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.
Factoring by Grouping
(9.8)
Polynomials and
Factoring
CHAPTER 9
THE
END