
Skills to Master
· Factor out a common monomial
· Factor by grouping
· Factor the difference of two squares
· Factor trinomials
· Know and use the properties of equality to solve equations
· Identify and solve linear equations
·
Solve for given
variables in formulas
Factoring Out a Common Monomial
If the terms of a
polynomial all have a common factor, that factor can be “removed” out in what
is basically the opposite of distribution.
Examples
![]()
![]()
![]()
![]()
Factoring By Grouping
In some cases, though
the terms of a polynomial may not have any common factors, pairs or more of
particular terms may be grouped together and each factored separately. Then each of the terms may again have factors
in common.
Examples
![]()
![]()
![]()
Factoring the Difference of Squares
If
a binomial happens to be a difference where each term is a perfect square, it
factors as the sum of the square roots of the terms times the difference of the
square roots of the terms.
Examples
![]()
![]()
![]()
Factoring Trinomials
Trinomials
that are the product of two binomials (i.e., formed by using FOIL) can be
factored. Generally, factoring trinomials is not easy. There exist various methods for doing
so. We’ll look first at an easy type of
trinomial to factor, then at a method for factoring more difficult trinomials.
Factoring a trinomial where the coefficient on the
square term is a 1
A
trinomial of the type we are considering has the following form:
, where a, b, and c are integers. We are
concerned here with trinomials where
.
To
factor such a trinomial:
1.
Find the factors
of the c term
2.
Choose the
factors that combine (by addition or subtraction) to give the b term
3.
Use these
factors as the second terms of each binomial
Examples
Factor
![]()
Factor
![]()
Factor
![]()
Factor
![]()
Factoring a trinomial where the coefficient on the
square term is not a 1
There
are various methods by which to factor a trinomial where the value of a is not 1. The method I’ll be showing you is called
“triple play” (you’ll see why).
To
use the triple play method:
1.
Find the factors
of the c term
2.
Choose the
factors that combine (by addition or subtraction) to give the b term
3.
Write down the a term three times as shown below
![]()
4.
Form two binomials by
placing the factors found in step two next to each ax on top (let’s say that the factors are m and n for illustration
purposes)
![]()
5.
Now one of two things
must be true:
a.
a divides evenly into
both terms of one of the two binomials
b.
a does not divide evenly
into either terms of the two binomials, but a
can be factored such that one of its factors divides into both terms of one of
the binomials, and the second factor divides into the two terms of the other
binomial
Examples
Let’s
look first at an example like part a. of number 5:
Factor
![]()
Now
let’s examine an example like part b. of number 5:
Factor
![]()
Try
these on your own:
Factor
(answer:
)
Factor
(answer:
)
Equations—Properties of Equality
An
equation is a statement indicating that two quantities are equal. If one or more variables are part of an
equation, then either there are one or more real numbers that cause the
equation to be true (in which case they are called the solutions to the equation), or there are no such numbers. When no such numbers exist, we say that the
equation has no solution or we can
say that the solution set is the empty set.
Our
goal in this section is to learn how to find the solution set (if it exists)
for an equation with a single variable.
Since the value of the variable is initially unknown, we will use properties of equality to form
successively simpler equivalent equations until we reach something like “x = “, where the solution is written in
the space to the right of the equal sign.
|
Addition/Subtraction
Properties of Equality |
Multiplication/Division
Properties of Equality |
|
If |
If |
|
|
|
|
|
|
Solving One-Step Equations
Examples
Solve
![]()
Solve
![]()
Solve
![]()
Solve
![]()
Solving Two-Step Equations
Examples
Solve
![]()
Solve
![]()
Solve
![]()
Solve
![]()
Solve
![]()
Solving Multi-step Equations
Examples
Solve ![]()
Solve ![]()
Solve ![]()
Solve ![]()
Solve ![]()
Solving Formulas
An equation that
contains two or more different
variables cannot be solved to produce a numerical value. But they can
be solved for individual variables using the same properties of equality.
Examples
Solve for C: ![]()
Solve for r: ![]()