Difference: DiagnosticTestAlgebra (1 vs. 2)

Revision 22007-12-10 - DickFurnas

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META TOPICPARENT name="DiagnosticTest"

Algebra Diagnostic Test

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 Combine and simplify expressions as much as possible.
Solve equations for x.
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  • Cannot be simplified.
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  • however, if we are told further that then we have a common factor in the denominator:
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  • however, if we are told further that then we have a common factor in the denominator:
    • , provided
 
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  • No algebraic simplification.
  • however... A beautiful geometric simplification is possible:
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    • is the hypotenuse of the right triangle with sides and
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    • is the hypotenuse of the right triangle with sides and
 
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Recall Pascal's Triangle:
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...in which elements in a row are the sum of the two elements in the row above it. These numbers are the Binomial Coefficients.

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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:
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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:
 
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which simplifies to:
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which simplifies to:
 
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  as previously.

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either or or
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  if a product is zero one of its factors must be zero so:
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  • Rearrange so the equation is of the form:
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  • and use the Quadratic Formula:
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Revision 12007-04-21 - DickFurnas

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META TOPICPARENT name="DiagnosticTest"

Algebra Diagnostic Test

Algebra

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

1.

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

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

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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.

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either or or

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

either or or

and either or or

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

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-- DickFurnas - 21 Apr 2007

 
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