Functions: Fractional functions
Long division with polynomials
If the degree of the numerator of a quotient function is greater than or equal to the degree of the denominator, we can write the quotient function as the sum of a quotient and a rest. The quotient is a polynomial and the remainder is again a quotient function.
Every function of the form
where and are polynomials and ,
can be rewritten using long division in the form
where is a polynomial and .
Example
gives
We use the following step-by-step guide for the division of polynomials using long division.
Long division
Step-by-step |
Example |
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For the quotient function we use long division |
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Step 1 |
Compose the long division as follows: |
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Step 2 |
Now find an such that if it is multiplied by , the term with the highest degree is equal to the term with the highest degree of . In the example on the right, we chose because is equal to . We put this term above the line in the long division. Now subtract the obtained expression from to get an expression . In the example, we found . Repeat this process until . |
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Step 3 | It now follows that
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Step 1 |
We first compose the long division: |
Step 2 |
Long division gives |
Step 3 |
According to the theory, it now follows that
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