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Algebra

Combine and simplify expressions as much as possible.
Solve equations for x.

1.

2.

3.

  • Cannot be simplified.
  • however, if we are told further that then we have a common factor in the denominator:
    • , provided

4.

5.

6.

7.

8.

  • No algebraic simplification.
  • however... A beautiful geometric simplification is possible:
    • is the hypotenuse of the right triangle with sides and

9.

Alternatively:

Recall Pascal's Triangle:

                1    1
             1    2    1
          1    3    3    1
       1    4     6    4     1
    1    5   10   10    5    1

...in which elements in a row are the sum of the two elements in the row above it. These numbers are the Binomial Coefficients. is the cube of the binomial so the coefficients can be taken from the third row above: 1  3  3  1 and the exponents on start with and count down, while the exponents on start with and count up:

which simplifies to:

as previously.

10.

11.

if a product is zero one of its factors must be zero so:

either or or

and either or or

12.

  • Rearrange so the equation is of the form:
  • and use the Quadratic Formula:

Trig

Answer the following questions without calculators or trig tables.
Leave answers like or as is.

1. Convert:

a. to radians.

b. to degrees.

c. to radians.

2. Evaluate:

a.

b.

c.

3. Sketch the graph of

(Make your vertical scale as large as possible.)

4. Given

5. Solve the following right triangle:

( i.e. determine missing sides and angles.)

a.

b.

c.

6. Relate to and

a.

b.

c.

7. Express in terms of and of and :

-- DickFurnas - 01 Mar 2007


This topic: MSC > DiagnosticTest
Topic revision: r10 - 2007-04-10 - DickFurnas
 
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