1 Quotient of Two Polynomials 5.6
By the end of this section, you should be able to solve
the fol lowing problems .
1. Find the quotient.
2. Find the quotient.
3. Use long division to find the quotient and remainder.
4. Use long division to find the quotient and remainder.
2 Concepts
In this treatment, we will divide a polynomial by a monomial by using a
technique called distributing the denominator . Many students believe that
they can safely omit this step, but to do so is to risk making the mistake of
cancellation across addition. Whenever students do this the error is always
the same, they cancel the denominator more times than is allowed for each
individual term in the numerator . We will not brook this fallacy. Therefore,
whenever we divide a polynomial by a monomial we must distribute the
denominator.
2.1 Example
Divide:
First we distribute the denominator, and then we do all
our cancellations
within terms.
+
3 Concepts
In our next example, we must divide a polynomial by a binomial. Here we
do long division until there is no more variable left . The algorithm is the
same as it was in grade school. First we divide, then we multiply, then we
subtract, then we bring down. We continue this process until the variable
has been divided out completely. In our next example, there is no remainder.
3.1 Example
Divide:
First we rewrite the problem in the following way.
Then we divide x2 by x, and we place the quotient over 5x
in the dividend.
Then we multiply x + 2 by x and subtract the product from x2 and 5x.
Notice that we add the opposite of x2+2x to x2+5x and bring down the
6. The next step is to divide 3x by x. We illustrate the rest of the problem
below.
Our next example contains a remainder.
3.2 Example
4 Facts
1. Whenever we divide a polynomial by a monomial, we must distribute
the denominator.
2. Whenever we divide a polynomial by a binomial we perform a four
step precess . (1) Divide by the divisor (2) Multiply the quotient by
the entire binomial (3) Subtract that product from dividend . (4) Bring
down the next term.
3. If there is a remainder, simply write that as a ratio added in at the end
of the quotient.
5 Exercises
1. Find the quotient.
2. Find the quotient.
3. Use long division to find the quotient and remainder.
4. Use long division to find the quotient and remainder.
6 Solutions
1. Find the quotient.
2. Find the quotient.
3. Use long division to find the quotient and remainder.
4. Use long division to find the quotient and remainder.
