Prime factorization, Greatest Common
Factor, and Least Common Multiple
10.1 Definition and simplifying
10.1.1 Definition
A rational expression is an expression that has a polynomial in the
numerator and denominator .
10.1.2 Example
An example of a rational expression is:
In this section we will learn to simplify rational
expressions. The best way to think
of a rational expression in as a fraction. When we simplify fractions, we break
down the
numerator and denominator into its prime factorization and then we cancel
factors common
to the top and bottom.
10.1.3 Example
Simplify:
Solution First , we start by using the skills we learned in
the last section to factor the
top and bottom to get:
10.1.4 Example
Simplify:
Solution We first factor the top and bottom and then
cancel:
10.2 Adding and Subtracting Rational Expressions
When adding and subtracting rational expressions, we need to first, find a
common denominator.
We’ll do this the same way we did it with fractions. We need to factor the
denominators into their prime factorization and then find the LCM of the
denominators :
10.2.1 Example
Add and simplify:
Solution We find the common denominator by factoring the
denominators completely
and the making sure that every factor is part of the common denominator. SInce
our
denominators are prime, we know that the common denominator is: (x − 3)(x + 1)
Notice
that both denominators are listed in the common denominator. The reason is is
that neither
divides evenly into the other.
So now we multiply top and bottom of each fraction by the missing factor of the
common
denominator:
10.2.2 Example
Subtract and simplify:
Solution First, we need to factor the denominators: x^2 −
x − 6 = (x − 3)(x + 2) and
x^2−2x−8 = (x−4)(x+2) So we have that the common denominator is:
(x−3)(x+2)(x−4).
Notice that we do not need to write the (x + 2) twice. As long as both
denominators divide
evenly into the common denominator, we are okay. Next, we multiply top and
bottom of
the fractions by the factor(s) needed to get a common denominator.
Notice that we were very careful with the minus sign
between the fractions. We need to be
sure the minus sign is distributed throughout the numerator of the second
fraction.
10.3 Multiplying and Dividing Rational Expressions
Again, we treat rational expressions as we would fractions. So, when multiplying
rational
expressions we factor the numerators and denominators, cancel factors common to
a top and
a bottom of one of the rational expression. When dividing rational expressions
we just flip
the second rational expression and then multiply.
10.3.1 Example
Multiply and simplify:
Solution First we factor the numerators and the
denominators to get:
10.3.2 Example
Divide and simplify:
Solution We, first, flip the second fraction to make a
multiplication problem. Then we
factor and cancel:
10.4 Complex Fraction
10.4.1 Definition
A complex fraction is a fraction with fractions in the numerator and/or
denominator.
We simplify a complex fraction by multiplying top and bottom by the common
denominator
of all the fractions both on top and on bottom. This method gets rid of all
fractions
except maybe the main fraction.
10.4.2 Example
Simplify:
Solution First, we need to factor all the denominators:
x^2 − 1 = (x − 1)(x + 1) and the
other two denominators are prime. SO our common denominator is: (x − 1)(x + 1).
So we
multiply top and bottom of the larger fraction by this:
10.4.3 Example
Simplify:
Solution FInd a common denominator first: (x − 2) and (x^2
+ 2x + 4 are both prime.
x^3 − 8 = (x − 2)(x^2 + 2x + 4). So the common denominator is: (x − 2)(x^2 + 2x
+ 4)
We multiply top and bottom by the common denominator to get:
Directions: Simplify the following:
Directions: Multiply or divide the following:
Directions: Simplify the following:
11 Equations
11.1 Linear Equations
Our goal when solving an equation is to undo order of operations. In many real
life situations
we take things apart by going through the put together steps in the opposite
order. The
same thing goes for solving equations. We do order of operations in the opposite
order.
11.1.1 Example
Solve for x: x + 2 = 5
Solution: We need to get x alone on one side. We start having 2 added to x. To
undo
+2 we need to subtract 2 from both sides. (We have to do the same thing to both
sides since
we are trying to keep the equality to remain true. If we subtract 2 from only
one side then
the two sides won’t be equal.)
So we have:
11.1.2 Example
Solve for t: 4t = 3
Solution Now we have t multiplied by 4. To undo multiplication by 4 we divide by
4:
So we have:
11.1.3 Example
Solve for w: 3w − 2 = 7
Solution Now we know by order of operations that the we first multiply w by 3
and
then we subtract 2 to simplify the left side. So we need to undo subtract 2 and
then undo
multiply by 3:
So we get:
We may be confronted with a more difficult looking
problem. If this is the case, we should,
first, simplify both sides then get the x’s on one side of the equation and then
undo order of
operations:
11.1.4 Example
Solve for x: 3(x − 4) + 2x = 2(2x − 5)
Solution So here we’ll simplify both sides and then we’ll move all x’s to the
left by
adding or subtracting by the correct multiple of x:
Finally, whenever we are confronted with an equation with
fractions, we should find the
common denominator of all the fractions and then multiply both sides of the
equation by
the common denominator.
11.1.5 Example
Solve for x: 
Solution We can see that the common denominator of all the fractions is 6. So we
will
multiply both sides by 6, then we’ll do as we have before to finish solving:
If we add absolute values to the equation then we need to
get rid of the absolute value
by splitting the equation into two equations.
11.1.6 Example
Solve for x: |x − 3| + 2 = 5
Solution First we need to get the absolute value by itself and then we write two
equations
one equal to positive and the other equal to negative the right hand side.
11.2 Quadratic Equations
There are different ways to solving quadratic equations. Some ways work better
for certain
forms of the equation. In this section we will go through all three ways.
The first way is called extracting square roots. In this method, we try to get
the squared
term by itself and then we undo the square and then get x by itself if
necessary:
11.2.1 Example
Solve for x: 2x^2 − 5 = 13
Solution We get x^2 by itself first and then we take the square root of both
sides:
Notice that I put ± in front of the square root. This is
because both a negative and a
positive square to get a positive.
11.2.2 Example
Solve for x: 2(x − 3)^2 + 2 = 8
Solution We’ll get (x − 3)^2 by itself, then we take square roots of both sides
and then
we’ll get x by itself:
The next way we can solve quadratic equations is to use
factoring: With this method,
we factor the quadratic and then set each factor to zero and solve the new
linear equations.
11.2.3 Example
Solve for x: x^2 + 6x = 7
Solution We need to first get the equation equal to zero and then we will finish
as stated
above:
The last way that we’ll discuss to solve quadratic
equations is to use the quadratic
formula:
Where we get a, b, and c are the coefficients of the
quadratic: ax^2 + bx + c
11.2.4 Example
Solve for x: 2x^2 − 5x + 3 = 0
Solution Using the formula and a = 2, b = −5, and c = 3 we get:
11.3 Linear Inequalities
Linear inequalities are solved in almost the same way as linear inequalites.
There is only one
main rule difference , that is, whenever we multiply or divide both sides of an
inequality by
a negative number, the inequality changes directions.
11.3.1 Example
Solve and graph the solution on a number line: −3x − 2 > 1
Solution
Section 11 exercises
Directions: Solve each linear equation:
Directions: Solve each quadratic equation by extracting
square roots:
Directions: Solve each quadratic equation by factoring:
Directions: Solve each quadratic equation using the
quadratic formula:
Directions: Solve the quadratic equations by any method.
Directions: Solve each inequality and graph the solution
on a number line:
