33 Zeros of Polynomial Functions
Learning Objectives
In this section, you will:
- Evaluate a polynomial using the Remainder Theorem.
- Use the Factor Theorem to solve a polynomial equation.
- Use the Rational Zero Theorem to find rational zeros.
- Find zeros of a polynomial function.
- Use the Linear Factorization Theorem to find polynomials with given zeros.
- Use Descartes’ Rule of Signs.
- Solve real-world applications of polynomial equations
A new bakery offers decorated sheet cakes for children’s birthday parties and other special occasions. The bakery wants the volume of a small cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. What should the dimensions of the cake pan be?
This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.
Evaluating a Polynomial Using the Remainder Theorem
In the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. If the polynomial is divided by[latex]\,x–k,\,[/latex]the remainder may be found quickly by evaluating the polynomial function at[latex]\,k,\,[/latex]that is,[latex]\,f\left(k\right)\,[/latex]Let’s walk through the proof of the theorem.
Recall that the Division Algorithm states that, given a polynomial dividend[latex]\,f\left(x\right)\,[/latex]and a non-zero polynomial divisor[latex]\,d\left(x\right)\,[/latex]where the degree of[latex]\,\,d\left(x\right)\,[/latex]is less than or equal to the degree of[latex]\,f\left(x\right)[/latex], there exist unique polynomials[latex]\,q\left(x\right)\,[/latex]and[latex]\,r\left(x\right)\,[/latex]such that
If the divisor,[latex]\,d\left(x\right),\,[/latex]is[latex]\,x-k,\,[/latex]this takes the form
Since the divisor[latex]\,x-k\,[/latex]is linear, the remainder will be a constant,[latex]\,r.\,[/latex]And, if we evaluate this for[latex]\,x=k,\,[/latex]we have
In other words,[latex]\,f\left(k\right)\,[/latex]is the remainder obtained by dividing[latex]\,f\left(x\right)\,[/latex]by[latex]\,x-k.\,[/latex]
The Remainder Theorem
If a polynomial[latex]\,f\left(x\right)\,[/latex]is divided by[latex]\,x-k,\,[/latex]then the remainder is the value[latex]\,f\left(k\right).\,[/latex]
How To
Given a polynomial function[latex]\,f,[/latex]evaluate[latex]\,f\left(x\right)\,[/latex]at[latex]\,x=k\,[/latex]using the Remainder Theorem.
- Use synthetic division to divide the polynomial by[latex]\,x-k.\,[/latex]
- The remainder is the value[latex]\,f\left(k\right).\,[/latex]
Using the Remainder Theorem to Evaluate a Polynomial
Use the Remainder Theorem to evaluate[latex]\,f\left(x\right)=6{x}^{4}-{x}^{3}-15{x}^{2}+2x-7\,[/latex] at[latex]\,x=2.\,[/latex]
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