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Study Guides > ALGEBRA / TRIG I

Algebra of Polynomial Functions

Learning Outcomes

  • Perform algebraic operations on polynomial functions
Just as we have performed algebraic operations on polynomials, we can do the same with polynomial functions. In this section, we will show you how to perform algebraic operations on polynomial functions and introduce related notation. The four basic operations on functions are adding, subtracting, multiplying, and dividing. The notation for these functions is as follows: Addition [latex](f + g)(x) = f(x)+ g(x)[/latex] Subtraction [latex](f − g)(x)= f(x) − g(x)[/latex] Multiplication [latex](f · g)(x)= f(x)g(x)[/latex] Division [latex]\frac{f}{ g} (x) = \frac{f(x)}{g(x)}[/latex] We will focus on applying these operations to polynomial functions in this section.

Add and Subtract Polynomial Functions

Adding and subtracting polynomial functions is the same as adding and subtracting polynomials. When you evaluate a sum or difference of functions, you can either evaluate first or perform the operation on the functions first as we will see. Our next examples describe the notation used to add and subtract polynomial functions.

Example

Let [latex]f(x)=2x^3-5x+3[/latex] and [latex]h(x)=x-5[/latex], Find the following: [latex-display](f+h)(x)[/latex] and [latex](h-f)(x)[/latex-display]

Answer: [latex-display]\begin{array}{lll}(f+h)(x)=f(x)+ h(x) & =(2x^3-5x+3)+(x-5) \\ & =2x^3-5x+3+x-5 & \text{combine like terms} \\ & =2x^3-4x-2 & \text{simplify}\end{array}[/latex-display] [latex-display]\begin{array}{lll}(h-f)(x)=h(x)-f(x) & =(x-5)-(2x^3-5x+3) \\ & =x-5-2x^3+5x-3 & \text{combine like terms} \\ & =-2x^3+6x-8 & \text{simplify}\end{array}[/latex-display]

In our next example, we will evaluate a sum of functions and show that you can get to the same result in one of two ways.

Example

Let [latex]f(x)=2x^3-5x+3[/latex] and [latex]h(x)=x-5[/latex] Evaluate: [latex](f+h)(2)[/latex] Show that you get the same result by 1) Evaluating the functions first then performing the indicated operation on the result. 2) Performing the operation on the functions first then evaluating the result.

Answer: 1) First, we will evaluate the functions separately: [latex-display]f(2)=2(2)^3-5(2)+3=16-10+3=9[/latex-display] [latex-display]h(2)=(2)-5=-3[/latex-display] Now we will perform the indicated operation using the results: [latex-display](f+h)(2)=f(2)+h(2)=9+(-3)=6[/latex-display]   2) We can get the same result by adding the functions first and then evaluating the result at [latex]x=2[/latex]. [latex](f+h)(x)=f(x)+h(x)=2x^3-4x-2[/latex] from the previous example. Now we can evaluate this result at [latex]x=2[/latex] [latex-display](f+h)(2)=2(2)^3-4(2)-2=16-8-2=6[/latex-display] Both methods give the same result, and both require about the same amount of work.

Multiply and Divide Polynomial Functions

We saw that multiplying polynomials often required the use of the distributive property and that the algebra of dividing polynomials could get messy fast!  The same techniques can be used to multiply and divide polynomial functions. Additionally, the same idea applies to evaluating a product or quotient of functions as we discovered in the previous example. We can either evaluate the function and then perform the indicated operation or vice-versa. You may already be thinking it will be a lot less work to evaluate the polynomials and then divide the results!

Example

Let [latex]g(t)=2t^3-t^2+7[/latex] and [latex]f(t)=5t^2-3[/latex] Find [latex](g · f)(t)[/latex], and evaluate [latex](g · f)(-1)[/latex]

Answer:

[latex]\begin{array}{lll}(g · f)(t) & =(2t^3-t^2+7)(5t^2-3) \\ & =(2t^3\cdot(5t^2)-t^2\cdot(5t^2)+7\cdot(5t^2))+(2t^3\cdot(-3)-t^2\cdot(-3)+7\cdot(-3))\,\,\,\,\,\text{apply the distributive property} \\ & =(10t^5-5t^4+35t^2)+(-6t^3+3t^2-21)\,\,\,\,\text{simplify}\\ & =10t^5-5t^4-6t^3+38t^2-21\,\,\,\,\,\text{combine like terms}\end{array}[/latex]

Evaluate [latex](g · f)(-1)[/latex] [latex-display]\begin{array}{lll}(g · f)(t)=10t^5-5t^4-6t^3+38t^2-21\\(g · f)(-1)=10(-1)^5-5(-1)^4-6(-1)^3+38(-1)^2-21=-10-5+6+38-21=8\end{array}[/latex-display]

In the next example, we will divide polynomial functions and then evaluate the new function.

Example

Given [latex]p(x)=2x^2+x-15[/latex] and [latex]r(x)=x+3[/latex] Find [latex]\frac{p}{r}(x)[/latex] and evaluate [latex]\frac{p}{r}(2)[/latex]

Answer: We can use synthetic division for this polynomial division since the coefficient on [latex]r(x)=x+3[/latex] is [latex]1[/latex]. Screen Shot 2016-07-15 at 4.09.44 PM This result means that [latex]\frac{p}{r}(x)=\frac{2x^2+x-15}{x+3}=2x-5[/latex] Now evaluate this quotient for [latex]x = -3[/latex] both ways as we did in a previous example. First, we will evaluate the result after polynomial division: [latex-display]\begin{array}{lll}\frac{p}{r}(x)=2x-5\\\frac{p}{r}(2)=2(2)-5=4-5=-1\end{array}[/latex-display] Next, we will evaluate each function for [latex]x = 2[/latex], then we will divide the results. [latex-display]p(2)=2(2)^2+(2)-15=8+2-15=-5[/latex-display] [latex-display]r(2)=(2)+3=5[/latex-display] Divide the results: [latex-display]\frac{p}{r}(2)=\frac{-5}{5}=-1[/latex-display]

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