How to Evaluate a Polynomial in Function Notation

Lesson Transcript
Instructor: Maria Airth

Maria has taught University level psychology and mathematics courses for over 20 years. They have a Doctorate in Education from Nova Southeastern University, a Master of Arts in Human Factors Psychology from George Mason University and a Bachelor of Arts in Psychology from Flagler College.

A polynomial is an algebraic expression that has more than one term and function notation is the way a function is written. In this lesson, explore how to evaluate or solve a polynomial in function notation. Updated: 10/07/2021

Definitions for Lesson

Hi, and welcome to this lesson on evaluating polynomials in function notation. Wow, that is a bit of a mouth full, don't you think? Let's break it down with some definitions first.

  • The prefix poly- refers to 'many,' and the term nomial means 'numbers' or 'terms.' So, a polynomial is many numbers or terms.
  • To evaluate means to solve in mathematical terms. You could think of it like working things out.
  • Finally, function notation is just the way a relationship is written to indicate how to evaluate a polynomial. A function is noted by using f(x), (f of x), notation, where the x is the argument of the function, or the value to use in the solution.

So, evaluating polynomials in function notation really means that we are going to solve functions of many terms by writing the method in a specific mathematical way.

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  • 1:18 An Analogy
  • 3:23 Evaluating Polynomials
  • 7:18 Lesson Summary
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An Analogy for Polynomial Evaluation

Are you hungry? Maybe we can take a break from talking about polynomials for a while and talk about baking. That sounds good. When you bake, say, a pie, you normally follow a recipe. In many recipes there is a special ingredient. Take this pie, the special ingredient is strawberries. It's a strawberry pie!

But, what if you don't like strawberries? Or you just want to try something different? You might think to yourself, 'I wonder what it would look like if I used blueberries instead?' Well, to use blueberries instead, all you would have to do is substitute blueberries in everywhere that the recipe says strawberries! Easy, no problem at all.

What does any of this have to do with the lesson? Thanks for asking.

What if I said that the strawberries were the X-tra special ingredient, and whenever you made a substitution, you were actually evaluating the x for the value of the substituted ingredient? What? Really? Yup, that is all there really is to it.

Remember that f(x) is how we indicate that something is a function, and it also gives the value of x (in the brackets) in order to evaluate the function. In our recipe, we could write: evaluate f(strawberry) when f(X-tra ingredient) = Recipe + X-tra ingredient. That means that after the evaluation, we would have the recipe with strawberries, or strawberry pie.

What about: evaluate f(blueberry) when f(X-tra) = Recipe + X-tra? You guessed it: blueberry pie.

Evaluating Polynomials with Function Notation

Do you see how evaluating the function really just means to substitute whatever is in the brackets of the 'f of' argument for the variable in the function equation? Let's look at an example:

Evaluate f(3) for f(x) = x + 3. This is a very simple function. It is stating that the answer will be 3 more than whatever value of x is chosen. The function notation tells us that we want to know what the answer will be if x is 3. To evaluate this function, we replace the x in the equation with 3 and solve: f(3) = (3) + 3 = 6. The solution is 6.

Now you try one: Evaluate f(4) when f(x) = 25 - x. I'll give you a little bit of time to do that.

Did you get 21? Great! When we substitute 4 in the place of the x, we have 25 - 4, and that is 21. Good job.

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