Download Chapter Nine: Polynomials and Factoring

Chapter Nine: Polynomials and Factoring
Section One: Adding and Subtracting Polynomials
In section 8.1 we discussed monomials. A monomial is an algebraic expression that is
either a number (constant), a variable, or a product of a number and one or more
variables. A polynomial is a monomial or the sum or difference of several monomials.
The degree of a polynomial with only one variable is equal to the highest power located
on the exponents. A polynomial in standard form is written with the powers in
descending order (highest to lowest).
Below are some examples of polynomials in standard form and their degree.
Degree 3: 4 x 3  3 x  5
Degree 2: 3 x 2  5 x  2
Degree 1: 2x 1
Degree 0: 5
EX1: Write each polynomial in standard form and give its degree.
a. 3  4 x 4  5 x  3 x 2
b. 9  3m2  m3  2m
Some polynomials have special names based on their degree or number of terms.
Degree
Name
Number of Terms
Name
0
constant
1
monomial
1
linear
2
binomial
2
quadratic
3
trinomial
3
cubic
4 or more
polynomial
4
quartic
5
quintic
EX2: Name the following polynomials.
a. 125  x3
b. x 2  5 x  6
We can add polynomials simply by dropping parenthesis and combining like terms. As a
reminder, like terms must have identical variables (the powers must be the same too).
EX3: Find the sum.
a.  3d 2  5d  1   4d 2  5d  2 
b.  3 x 2  5 x    4  6 x  2 x 2 
Subtraction is achieved by using opposites. Reminder: when a negative is in front of
parenthesis, we drop parenthesis and change all signs inside it.
EX4: Find the difference.
a. 10 p 2  5    7 p 2  2 p  3
b.  3x 2  2 x  3   x 2  2 x  4 
Section Three: Multiplying Polynomials
Multiplying two binomials is simply an extension of the distributive property.
EX1: Multiply.
a. 5x  2 x  1
b. 3x  2 x  5
The method for multiplying two binomial is sometimes called FOIL. The word FOIL tells
you what all you have to multiply together to find the product of the two binomials: First,
Outer, Inner, and Last. We can simply just remember to multiply every term by every
term.
EX2: Use FOIL to multiply.
a.  2 x  1 5x  3
b.  4 g  7  2 g  3
c.  3m  5 3m  5
d.  3x  5 4 x  2 
There are two types of special products. When we multiply conjugates (the only
difference is the plus and minus), look what happens.
EX3: Multiply.
a.  x  3 x  3
b.  2 x  1 2 x 1
The other type of special product occurs when we square a binomial.
EX4: Multiply.
2
a.  x  5 
b.  2 x  6 
c.  x  3
2
2
d.  3 x  4 
2
EX5: A rectangular poster is twice as long as it is wide. It is mounted on a piece of
cardboard with a 3-inch border on all sides. The area of the border is 306 square inches.
What are the length and width of the poster?
Section Five: Common Factors
Factoring is the process of taking an expression and breaking it into it factors. Factoring
is the opposite of multiplying. A simple example is shown below.
Multiplying: 2  3  6
Factoring: 6  2  3
The first things that we will factor will simply be the distributive property in reverse.
Multiplying: 4 x  2 x  3  8x2  12 x
Factoring: 8x2  12 x  4 x  2 x  3
The steps of factoring with the distributive property:
1. Find the greatest factor that is common to all the terms
2. Write this term outside of parenthesis
3. What could you multiply this term by to give you the original expression
EX1: Factor each polynomial.
a. 4 y 3  16 y 4
b. 6 p  15 p 2  9 pq
c. 8r 4  17 s 4
Sometimes the common factor could be a little “uglier.”
EX2: Factor.
a. a  b  4  c  b  4 (There are only two terms)
b. a  x  3  6  x  3
Factoring by grouping will be extremely important as we continue the chapter on
factoring.
EX3: Factor by grouping.
a. ax  bx  ay  by
b. ax  bx  ay  by
c. mn  mp  5n  5 p
d. 3x 2  3xy  2 xy  2 y 2
Section Six: Factoring Special Polynomials
EX1: Factor the quadratic trinomials using the methods from section seven.
a. x 2  8 x  16
b. x 2  10 x  25
c. x 2  6 x  9
d. x 2  12 x  36
These trinomials are called perfect square trinomials because they factor into perfect
squares. We can look at a trinomial before we factor and tell if it is a perfect square. Let’s
see if 4 x 2  12 x  9 is a perfect square trinomial by using the following test:
1. Is the first term a perfect square? Yes,  2 x  2 x 
2. Is the last term a perfect square? Yes,  3 3
3. Is half the middle term, 6x, the product of the square roots of the first and last
terms? Yes,  2 x  3  6 x
EX2: Are the following perfect-square trinomials?
a. 25 x 2  30 x  3
b. x 2  12 xy  36 y 2
c. 4 x 2  18 x  81
d. 100m2  140m  49
Another polynomial that appears quite often is the difference of squares: x 2  16 . Notice
that both the first and second terms are perfect squares. Also, the two squares are being
subtracted. That is why this is called difference of squares. From section three we know
that this will factor into conjugates  x  4 x  4 .
EX3: Are the following difference of squares?
a. 4 y 2  49
b. 64  5a 4
c. 81g 2  16
EX4: Factor each difference of squares.
a. m 2  121
b. 9 x 2  100 y 2
c. 25x 2 y 4  36r 2 s 6
Section Seven: Factoring Quadratic Trinomials
Many, but not all, quadratic trinomials will factor into the product of two binomials. The
steps for factoring these trinomials are as follows:
1. Make sure your quadratic is in standard form:
2. Factor out any common factors (this cleans up and makes things easier)
3. Draw your two sets of parenthesis
4. The two things that multiply to give you your first term are the first part of each
term
5. Ask two questions. If ax 2  bx  c ,
a. What two numbers add to give us our b?
b. What same two numbers multiply to give us our c?
6. This is the second part of each term
EX1: Factor the trinomials.
a. x 2  12 x  35
b. x 2  14 x  40
c. x 2  10 x  16
d. x 2  12 x  28
e. x 2  4 x  21
f. x 2  x  20
When there is a number in front of the x 2 term, we must factor by grouping. Ask
yourself the same two questions from before. When you get your two numbers, split your
b term into these two numbers. Then factor by grouping.
EX2: Factor the trinomials.
a. 3x 2  10 x  3
b. 3x 2  4 x  15
c. 2 x 2  7 x  5
d. 6 x 2  7 x  3
EX3: Factor completely. (Factor out the common term first)
a. 3 x3  27 x 2  42 x
b. 2 x 3  9 x 2  9 x
Section Eight: Solving Equations by Factoring
We can solve quadratic equations by using factoring. If two things are being multiplied
and give you zero. Then we know that one of those things have to be a zero as well. This
is called the zero product property.
EX1: Solve the equation using the zero product property.
a.  x  3 x  2  0
b.  2 x  5 3x  4  0
Using factoring, we can use this same methods to solve other equations.
EX2: Factor out a common factor to solve the equation.
a. 8 x 2  40 x  0
b. 10 x 2  100 x  0
EX3: Solve each equation.
a. x 2  12 x  27  0
b. x 2  6 x  16  0
c. x 2  5 x  6  0
d. 2 x 2  5 x  3  0
We can use this principle to find the x-intercepts of a quadratic function. Remember that
to find the x-intercepts, you plug in zero for y and solve. Another name for the xintercepts are the zeros or the roots.
EX4: Find the x-intercepts of the following graphs. Check with a graph.
a. y  ( x  3)( x  4)
b. y  x 2  3x  10
c. y  x 2  9 x  18
d. y  2 x 2  9 x  9