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Dividing Polynomials

Dividing Polynomials if the denominator is a monomial .

 We add and subtract fractions with a common denominator using the fol lowing rule . If there is a
common denominator then combine the numerators and put them over that common denominator.

 This process could be written backwards and still be a true equation .

 This rule states that if we have a polynomial with several terms in the numerator of a fraction then we
can break that fraction into several separate fractions. Each of the separate fractions will have one term
of the polynomial in the numerator and the common denominator in the denominator.

Why break up the fraction with a monomial in the denominator ?

 The quotient rule can be used to reduce fractions but the numerator and denominator must both be
monomials. The quotient rule cannot be used if the numerator or denominator has more that one term.
We CANNOT cancel parts of the polynomial.

You cannot reduce the parts
of addition or subtraction
You can reduce if everything
is a factor
the 4 and the 8
and x the and the x^2
cannot be reduced
the 4 and the 8
and x the and the x^2
CAN be reduced

 If the denominator of the fraction is a monomial then you break up each fraction into its separate parts
and then each separate fraction can be reduced using the quotient rule.

Example 1 Example 2 Example 3

Dividing Polynomials if the denominator is a Binomial: Long Division
Why breaking up the fraction with a binomial in the denominator will not work.

 Breaking up a fraction into its separate parts and using the quotient rule to reduce each separate
fraction will not work if the denominator is a binomial. If we did break up the fraction then each separate
fraction would have the binomial as their denominator.

 The quotient rule cannot be used to reduce each separate fraction if the numerator and denominator are
not both monomials. We must develop another process to reduce the fraction. The process we use
is to divide the binomial in the denominator into the numerator by using long division. This long
division process will look like the long division process for fractions with whole numbers that you have
used
in the past. The process will be a bit more involved due to the nature of the algebraic terms
involved.

Long division with whole numbers

 divide the 2 outside into the first number inside (9) and put the results over the 9

multiply the 4 on top by the 2 outside and write the product 8 under the 9

multiply the 8 in the new row by −1 to create a subtraction problem

combine the terms above the under line and bring down the 7

Start the process over again

 Divide 2 into 17 to get 8. Put the 8 above the 7 and multiply the 2 and 8. Put the 16 under the 17 multiply the 16 in the new row by −1 to create a subtraction problem We are done and have a remainder of 1 the answer is

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