Real Zeros of Polynomial Functions

Long Division of Polynomials

You can use long division of polynomials to find the real zeros of polynomial functions. For example:

Divide  by x-2.

 

Steps:

1.      Multiply (x-2) by 6x² to get 6x³. 

2.      Subtract.

3.      Multiply (x-2) by -7x to get -7x².

4.      Subtract.

5.      Multiply (x-2) by 2 to get 2x-4

6.      Subtract

You can use this solution to factor the function completely.  This is true by the remainder theorem, which states that if a polynomial f(x) is divided by x-k, the remainder is r=f(k).  In other words, if the remainder is zero, it as a factor and follows the factor theorem, which states that a polynomial f(x) has a factor (x-k) if and only if f(k)=0.

You can use this factorization to determine the zeroes of the polynomial function.

As you can see, the zeros of the function occur at 2, ½, and ²/₃.

Now what happens if you have a remainder other than zero?

If you have a remainder that is other than zero, you divide the remainder by the divisor and add it into the quotient.  For example:

This illustrates the difference algorithm which states that the dividend equals the divisor times the quotient times the remainder.

Important note – always make sure to use placeholders when there are missing terms in the dividend.  For example:

Synthetic Division

Synthetic division is a nice shortcut for long division; however keep in mind that it works only for dividing polynomials by divisors of the form x-k.  Here is an example:

Divide

Divideby x+3

Steps:

1.      Set up the array. Make sure to include placeholders when there are missing terms in the dividend.

2.      Drop the first number below the line and multiply it by the number in the upper left corner.

3.      Take the solution and put it above the line one place to the right.

4.      Add the two numbers above the line and write the product below the line.

5.      Repeat steps 2-5 until you have reached the final number

6.      If the final number is zero, there is no remainder.  If the final number is not zero, divide the remainder by the divisor and add it to the quotient.

A diagram of how the array is set up is shown below.

Rational Zero Test

The Rational Zero test relates the possible rational zeroes of a polynomial (having integer coefficients) to the leading coefficient and to the constant term of the polynomial.   The technical definition is if the polynomial

 has integer coefficients, every rational zero of ƒ has the form Rational zero=p/q where p and 1 have no common factors other than 2, p is a factor of the constant term a⁰,  and q is a factor of the leading coefficient aⁿ.  The rational zero test essentially tells us that the possible rational roots are the factors of the constant term divided by the factors of the leading coefficient. Here is an example of how to use the rational zero test to solve for the rational zeros of a polynomial: 

Normally when trying to solve for rational zeroes, you randomly plug in the possible rational zeroes and keep trying until one works.  This can be extremely lengthy and tedious.  Instead, if you graph the function on a graphing calculator, you will be able to calculate its zeroes by using the CALC function.  If the x-value is one of the possible rational roots, plug it into the upper left corner of the synthetic division array.  It will result in a zero remainder, meaning it factors cleanly and allows you to get onto the next step for synthetic division.  In addition, whenever the coefficients of a function add up to zero, (x-1) will always equal zero, meaning 1 is a zero of the function.

Works Cited

"Precalculus with Limits: A Graphing Approach." Precalculus with Limits: A Graphing Approach. N.p., n.d. Web. 15 Jan. 2015.