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Lesson 8: Dividing polynomials

Topic. This lesson covers Session 8: Dividing polynomials

Learning Outcomes.

  • 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:

  1. 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:

    \[5, \quad x, \quad 7 x^{2} y, \quad-12 x^{3} y^{2} z^{4}, \quad \sqrt{2} \cdot a^{3} n^{2} x y\]

The following are examples of polynomials:

    \[x^{2}+3 x-7, \quad 4 x^{2} y^{3}+2 x+z^{3}+4 m n^{2}, \quad-5 x^{3}-x^{2}-4 x-9, \quad 5 x^{2} y^{4}\]


In particular, every monomial is also a polynomial.

In this class, we will focus on polynomials with just one variable x — we will consider these as functions. For example f(x)=x^2+3x-7 is such a polynomial.

Definition. A polynomial is a function f of the form

    \[f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{2} x^{2}+a_{1} x+a_{0}\]

  • The real numbers a_{0}, a_{1}, \ldots, a_{n} are called coefficients (sometimes we allow the coefficients to be complex numbers as well).
  • The domain of a polynomial f is all real numbers (sometimes we allow the domain to include complex numbers as well).
  • The coefficient a_{n} in front of the highest power of x is called the leading coefficient and n is the degree of the polynomial.
  • The zeros of a polynomial are usually referred to as roots. Therefore x is a root of a polynomial f precisely when f(x)=0.

Definition. A rational function is a fraction of two polynomials f(x)= \frac{g(x)}{h(x)}, where g(x) and h(x) are both polynomials. The domain of f is all real numbers for which the denominator h(x) is not zero:

    \[D_{f}=\{x \mid h(x) \neq 0\}\]

Example. The following are examples of rational functions:

    \[f(x)=\frac{-3 x^{2}+7 x-5}{2 x^{3}+4 x^{2}+3 x+1}, \quad f(x)=\frac{1}{x}, \quad f(x)=-x^{2}+3 x+5\]

Long Division

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:
a) \frac{3571}{11}
b) \frac{x^{3}+5 x^{2}+4 x+2}{x+3}

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:

    \[3571 = 324\cdot 11 + 7 \text{, or } \frac{3571}{11}=324+\frac{7}{11}\]

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 x^{3}+5 x^{2}+4 x+2 by x+3 we get a result of x^2+2x-2 and a remainder of 8. We can express this in two different ways:

    \[x^{3}+5 x^{2}+4 x+2=\left(x^{2}+2 x-2\right) \cdot(x+3)+8 \text{ or }\]

    \[\frac{x^{3}+5 x^{2}+4 x+2}{x+3}=x^{2}+2 x-2+\frac{8}{x+3}\]

In polynomial division problems, we use the same names for each item — but in this case, they are functions instead of numbers:

  • f(x)=x^{3}+5 x^{2}+4 x+2 is the dividend (the thing we are dividing into)
  • g(x)=x+3 is the divisor (the thing we are dividing by)
  • q(x)=x^{2}+2 x-2 is the quotient (the number of times it goes in)
  • r(x)=8 is the remainder (whatever is left). NOTE: The divisor will always be a polynomial with degree less than the degree of the divisor.

Dividing by (x-c)

We now turn our attention to a particular example of division, where the divisor has the form g(x)=x-c. Why do we care about this form in particular? It turns out that dividing by x-c has a nice connection to the roots, or zeros, of a polynomial.

Observation. Suppose we divide a polynomial f(x) by a polynomial g(x)=x-c, and the result has remainder r:

    \[f(x)=g(x)\cdot (x-c)+r\]

Now, let us plug the value x=c into both sides of this equation:

    \[f(c)=g(c)\cdot(c-c)+r = g(c)\cdot 0 +r = r\]

Notice that the remainder r is exactly equal to what we get when we evaluate the function f at c (we call this fact the remainder theorem):

The reminder when dividing f(x) by (x-c) is r=f(c).

Thinking about what happens when we get a remainder of 0, we come up with the factor theorem — it tells us that the polynomial (x-c) will go into f(x) evenly (no remainder) exactly when f(c)=0.

    \[f(c)=0 \text{ exactly when }g(x)=x-c\text{ is a factor of }f(x)\]

Example.
a) Find the remainder of dividing f(x)=x^2+3x+2 by x-3
b) Determine whether g(x)=x+3 is a factor of f(x)=x^4+4x^3+x^2+18
c) Show that 5 is a root of f(x)=x^3-19x-30, and use this to factor f(x) completely.

VIDEO: Factor Theorem and Remainder Theorem examples

Exit Question

Divide by long division \frac{2 x^{4}+7 x^{3}+x+3}{x+5} and use the result to rewrite f(x)=2 x^{4}+7 x^{3}+x+3

Answer

f(x)=2 x^{4}+7 x^{3}+x+3=2 x^3 - 3 x^2 + 15 x - 74+\frac{373}/{x + 5}