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Intermediate Algebra
Spring Semester (S2) Notes

Chapter 6 - "Rational Expressions"

Factor quadratic trinomials using diamond problems.
Then cancel common binomial factors.

Worksheet--What Do You Call an Alligator That Sneaks Up and Bites You From Behind?
#1-5

Also, use the difference of two squares formula:  (a-b)(a+b)
#6-8

Find greatest common factors (GCF).
#9-10

#11-13

#14-15
For #14, multiply the leading coefficient by the constant term,
then do a diamond problem with -4 and +3.  Factor by grouping (or box method.)

ATAILGATOR

What do you do when you have 3-x instead of x-3?  Factor out a "-1".
(3-x) = -1(x-3).  This means that (3-x) / (x-3) reduces to -1.
Is that true?

(3-5) / (5-3) = -2 / 2 = -1 and
(7-4) / (4-7) = 3/(-3) = ?.
This is true for all numbers.

PZ108 "Cryptic Quiz"
1. What do you call a skydiver with the flu?

(9-x^2) = (3-x)(3+x)
-(x^2-9) = -1 (x-3)(x+3)

2. How do you crash a houseboat party?

Simplifying Rational Expressions

PZ109 "What do you call an insect that plays drums?"

A RHYTHMIC TICK

BOOKS NEVER WRITTEN

Multiplying Rational Expressions

What Do You Call a Message Printed on a Lion With Chickenpox?

PZ112 Why Are Ancient Stories Like Feet?

(Match the work below with its problem and answer.)

THEY ARE BOTH LEG ENDS

Dividing Rational Expressions

To divide rational expressions, flip the second fraction to multiply by the reciprocal.

Then factor and cancel common factors.
Note the difference of squares a^2 - b^2 = (a - b)(a + b).
(There is a mistake in the photo.  The corrected solution is below.)

Be careful, common factors in the numerator do not cancel.

The next problem has a difference of cubes factorization:
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
#67 involves factor by grouping:
ab - 2b +3a - 6 =
b(a - 2) + 3(a - 2) =
(a - 2) (b + 3)

The denominator in #68 involves factor by grouping.

What Lives in the Sea and Yells?

A CLAM SHOUTER

Why Did Orgo Take a Bath After Walking Through Mudsucker Swamp?
Find common monomial denominators.

The numerators are:
#6 5a(a+4) + 3(2a-1) =
#7 3(a+6) + a(4a+3) =
#8 3a^2(a-4) +6a(1) + 2(7-3a) =
#9 a(3a+b) + b(5a-2b) =
#10 b(2a+2) + 7(b-9) =

Finding common polynomial denominators.
The work to find each common denominator is shown below:

Match the work with its problem and answer on this worksheet.

Match the work in the photo gallery above with each problem and answer below.

Complex Fractions

Fractions imply division.

You can also multiply top and bottom by the same quantity to remove denominators.

p389#1

To divide a fraction, multiply by its reciprocal.

p389#3

Find common denominators in the numerator and denominator.

p389#5
Cancel like algebraic expressions.

p.389#7
Find common denominators, the reciprocal, then distribute.
Why can't 9x^2 terms cancel?

p389 #9
Find common denominator, then multiply by reciprocal.

p389 #11
Multiply numerator by denominator's reciprocal.  Can the 2x's cancel?

p389 #13
Multiplying top and bottom by x^2 cancels each denominator.

p389 #15

Note the difference of cubes factor.

Polynomial Long Division

Rewrite this expression with the long division symbol:

Here's an animated gif on how to complete the problem.

Here is each step of the problem:

1) Setup the problem.      2)  What is 2x^2/x?      3)  Multiply 2x(x)      4)  Multiply 2x(5).

5) 2x(x+5) = ?      6)  Subtract (2x^2+10x)      7) by reversing signs,      8) add to get -3x.

9) Bring down -15.      10) -3x/x = -3      11) multiply -3(x)      12) and -3(5)

13) write -3(x+5) product.      14) Subtract -3x-15,      15) or add 3x+15      16) to get 0.

The remainder is zero.

Here's the final solution.

What if there is a remainder?

Divide

There is a remainder of 5.

Here's how to do #23 step by step:

Divide this quartic by this binomial.

Account for 0 x-squareds when writing coefficients.

Now divide    using synthetic division:
Solve x - 4 = 0 first to get the divisor's root.
Then write the coefficients in descending order.

The remainder of 304 is written as a fraction: .

This next example is illustrated with a gif:

Here's how to proceed:

The remainder is 19:

SOLVING RATIONAL EQUATIONS

PZ131 E

pz131 G
pz131 G

Rational Exponents

are like radicals because they have the same index (root number which is 3) and the same radicand (number under the radical which is 5.

are not like radicals because they have different radicands 8 and 9.

are like radicals because they have the same index (2 for square root) and the same radicand 2 x.

Examples

Simplify the following expressions

Solutions to Above Examples

The above expressions are simplified by first factoring out the like radicals and then adding/subtracting.
More Examples

Simplify the following expressions

Solutions to Above Examples

The above expressions are simplified by first transforming the unlike radicals to like radicals and then adding/subtracting

When it is not obvious to obtain a common radicand from 2 different radicands, decompose them into prime numbers. Decompose 12 and 108 into prime factors as follows.

We now substitute 12 and 108 by their prime factors and simplify

Questions With Solutions

Simplify the following expressions
Solutions

3. The 3 radicands in the given expression -√ 32 - 2√ 50 + 3√ 200 are different but note that 32, 50 and 200 may be written as 2 times a number that is a perfect square as follows: 32=2 * 16, 50=2 * 25 and 100=2 * 100. Substitute in the given expression and simplify.

Decompose 28 and 63 into prime factors as follows: 28=2 2 * 7 , 63=3 2 * 7 and substitute into the given expression and simplify

7.

Completing the ...
SQUARE!

x^2 + 4x + ?

x^2 + 6x + ?

(x+4)^2 =

Click on pic below for active demonstration:

Solve x^2 + 6x + 8 = 0

with sound effects (lasers, whips, and explosions!)

Use the formula to solve:

Use the formula to solve:

Use the formula to solve:

vertex = -b / (2a)

Window for #45                                                   Window for #50

Two numbers add to 11:  x + y = 11.   Their product xy is 11.
Solve for y:  y = 11 - x.   Then product is x(11-x).
Graph and find vertex for maximum product.
-b/2a = -11/(2*-1) = 5.5

Click PZ223 Calculator for the rest of the keystroke solutions.

Composition of Functions
(fog and gof)

Fill in the table below: