Factoring and simplifying

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Factoring and simplifying

• Review Chapter

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Worksheets

• Factors versus terms will investigate the
  important distinction between terms and factors
  in equations. It helps to explain solving
  techniques.
• Review exercises College Algebra is a review
  of material such as that in the review chapter of
  your book. Additional problems can be found
  there.
• A key to Review exercises College Algebra is
  available.
• Factoring trinomials discusses a few ways to
  factor trinomials. Use whichever way best suits
  you.

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Worksheets

• Scientific notation and your calculator
  explains scientific notation and how your
  calculator uses it.
• Properties of real numbers goes through the
  basic properties we will use throughout the
  semester. We work with actual numbers, and then
  expand our knowledge to the algebraic notation.
  This will help us understand simplifying
  expressions.
• PEMDAS explains and provides practice problems
  for the order of operations. It is crucial that
  you understand the order in which to perform
  operations in arithmetic and algebraic
  expressions.

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Factors versus terms

• Many people like to think that they can cancel
  out something if
• its on the top and the bottom like the 2 in
  .

To see that it does not work to cancel the
2s to get , put 5 in for in
. You get
The 2 does not cancel. We see this because if we
put 5 in for x we see that
5
What can you cancel?

• Now cancel the 3 from to get .
• Does this work? Substitute 5 in for x for both
  expressions and see if they are equal.

Theyre equal! Why did it work this time and not
before?
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When can you cancel?

• The difference between and is that the 3
• is a factor of both the top and bottom,
  meaning it is multiplied by everything else, its
  3 times (x 4) on top and 3 times x on bottom.
  So watch what happens.

Because of how fractions multiply, you can break
it apart to reveal a separate factor
of
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Why can you cancel?

• The difference was that 3 was a common factor.
  The 2 in
• is a term on the top. It is x minus 2. The
  operation
• is addition (or subtraction), not
  multiplication (or division). You cannot cancel
  out a number if its a term and not a factor.
  Lets try another! Check both equations by
  substituting a value for x.

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Let x 2.
Let x 2.
Nope!
It works!
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Why did it work?
Notice there is a common factor between the terms
on top. Factor it out. Then cancel it with the
bottom factor of 5.
So you can only cancel something from top and
bottom if it is a factor of both the top and
bottom.
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Rules of exponents

• Work through the rules of exponents. They will
  stick better to memory.

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Simplifying
b to the negative 4 on top is b to the
positive 4 on bottom
Deal with inside first.

• Simplify

We have and on bottom
Just square the fraction.
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Simplifying
Factor the top. Then multiply the fractions.

• Simplify

Here weve got common factors of 2 and (x 1) on
top and bottom.
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Simplifying
The same rules apply.
Prepare exponents for addition or subtraction.

• Simplify

Leave exponents positive in answer.
Prepare exponents for subtraction.
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Simplifying
First, simplify the top and bottom separately.

• Simplify

and
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Simplifying
The circled factors are not perfect squares.
Theyll stay inside the radical.
Factor perfect squares out.

• Simplify

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Simplifying
Factor out perfect cubes.
The circled factors are not perfect cubes.
Theyll stay inside the radical.

• Simplify

and is the number I cube to get 27, or 3.
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Try these problems to test your review knowledge.