Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Lesson 8: Dividing polynomials
Topic. This lesson covers Session 8: Dividing polynomials
- Use the long division algorithm to divide two polynomials, determining the quotient and remainder,
- Understand the connection between long division, factors, and roots, and use this connection to solve problems.
WeBWorK. There is one WeBWorK assignments on today’s material:
- Polynomials – Division
Additional Video Resources.
Question of the Day: When we divide two numbers, the result is a number. When we divide two polynomials, what is the result?
Introduction to Polynomials
Definition. A monomial is a number, a variable, or a product of numbers and variables. A polynomial is a sum (or difference) of monomials.
Example. The following are examples of monomials:
The following are examples of polynomials:
In particular, every monomial is also a polynomial.
In this class, we will focus on polynomials with just one variable — we will consider these as functions. For example is such a polynomial.
Definition. A polynomial is a function of the form
- The real numbers are called coefficients (sometimes we allow the coefficients to be complex numbers as well).
- The domain of a polynomial is all real numbers (sometimes we allow the domain to include complex numbers as well).
- The coefficient in front of the highest power of x is called the leading coefficient and is the degree of the polynomial.
- The zeros of a polynomial are usually referred to as roots. Therefore is a root of a polynomial precisely when .
Definition. A rational function is a fraction of two polynomials where and are both polynomials. The domain of is all real numbers for which the denominator is not zero:
Example. The following are examples of rational functions:
We now show how to divide two polynomials. The method is very similar to the long division of natural numbers.
Example 8.6. Divide the following fractions via long division:
VIDEO: Example – Long division of Natural Numbers and Polynomials
When we divide 3571 by 11 we get a result of 324 remainder 7. We can express this in two different ways:
In division problems, we have names for all these numbers:
- 3571 is the dividend (the thing we are dividing into)
- 11 is the divisor (the thing we are dividing by)
- 324 is the quotient (the number of times it goes in)
- 7 is the remainder (whatever is left)
When we divide by we get a result of and a remainder of . We can express this in two different ways:
In polynomial division problems, we use the same names for each item — but in this case, they are functions instead of numbers:
- is the dividend (the thing we are dividing into)
- is the divisor (the thing we are dividing by)
- is the quotient (the number of times it goes in)
- is the remainder (whatever is left). NOTE: The divisor will always be a polynomial with degree less than the degree of the divisor.
We now turn our attention to a particular example of division, where the divisor has the form . Why do we care about this form in particular? It turns out that dividing by has a nice connection to the roots, or zeros, of a polynomial.
Observation. Suppose we divide a polynomial by a polynomial , and the result has remainder :
Now, let us plug the value into both sides of this equation:
Notice that the remainder is exactly equal to what we get when we evaluate the function at (we call this fact the remainder theorem):
The reminder when dividing by is .
Thinking about what happens when we get a remainder of , we come up with the factor theorem — it tells us that the polynomial will go into evenly (no remainder) exactly when .
a) Find the remainder of dividing by
b) Determine whether is a factor of
c) Show that 5 is a root of , and use this to factor completely.
VIDEO: Factor Theorem and Remainder Theorem examples