When working with polynomial functions, you might be asked to evaluate them for a specific value of the variable. Do you know what to do in this case? Also, according to their degree, polynomial functions will have different types of graphs. Do you know how to solve polynomials by graphing and how to recognise the different types of polynomial graphs? In this article, we will answer all these questions using practical examples.
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Jetzt kostenlos anmeldenWhen working with polynomial functions, you might be asked to evaluate them for a specific value of the variable. Do you know what to do in this case? Also, according to their degree, polynomial functions will have different types of graphs. Do you know how to solve polynomials by graphing and how to recognise the different types of polynomial graphs? In this article, we will answer all these questions using practical examples.
Firstly, let's recall what do we mean by polynomials.
Polynomials are expressions involving multiple terms that contain a variable raised to a series of positive whole-number exponents. Each term may also be multiplied by coefficients.
Polynomial functions are functions that follow the standard form:
As you can see, polynomial functions in standard form are written in decreasing order, from the biggest exponent to the smallest. Also from the equation above, you can notice the following characteristics of a polynomial:
is the leading term of the polynomial, since this is the term with the highest exponent.
is the leading coefficient, because this is the coefficient of the leading term of the polynomial.
The highest power or exponent present in a polynomial is called the degree of the polynomial. In this case, the degree is n.
is a constant because this term is not accompanied by a variable.
is a polynomial of degree 2
is not a polynomial because it has a negative exponent
There are a couple of methods that you can use to evaluate polynomials, which are direct substitution and synthetic substitution.
To evaluate a polynomial using direct substitution, all you need to do is substitute x with a number to find its solution.
Evaluate when using direct substitution
Using synthetic substitution, the process is different. Let's evaluate the polynomial function from the previous example to explain the steps that you need to follow using this alternative method.
Evaluate when using synthetic substitution
The steps to follow with synthetic substitution are:
Step 1. Write down the coefficients of each term in the polynomial function in descending order. To the left-hand side of the leading coefficient, write down the value of x that you are evaluating the polynomial function with, as shown below.
Remember to add any missing terms with a coefficient of zero (0).
Step 2. Bring down the leading coefficient below the horizontal line. Multiply the leading coefficient by the value of x. Write down the result of the multiplication just under the following coefficient. Then, add the values in the second column taking into account their signs. Write down the result of the addition directly below the horizontal line.
Bring down the leading coefficient, which is 3
Multiply the leading coefficient times the value of x: , and put the result under the following coefficient
Add the values in the second column:
Step 3. Multiply the result of the addition by the value of x, and put the result of the multiplication just under the following coefficient. Then, add the values taking into account their signs. Repeat this step for all the coefficients. The final value below the horizontal line will be the value of , in this case, .
Multiply 14 times the value of x: , and put the result under the following coefficient
Add the values in the third column:
We do not need to repeat this step, because there are no more coefficients left.
Notice that this is the same result that we obtained using the previous method.Polynomial graphs are graphical representations of polynomial functions.
There are different types of polynomial graphs according to their degree:
Notice that the degree of a polynomial matches the number of direction changes in their graph and the number of zeros or x-intercepts.
Degree 1 - Linear | Degree 2 - Quadratic |
Degree 3 - Cubic | Degree 4 - Quartic |
Degree 5 - Quintic | Degree 6 -Hexic |
As the behaviour of higher exponent functions is not as predictable as lines or parabolas, to get a more accurate representation of their curve, we need to use some key features.
1. Find the zeros: The zeros of a function are the values of x that make the function equal to zero. They are also known as x-intercepts.
To find the zeros of a function, you need to set the function equal to zero and use whatever method required (factoring, division of polynomials, completing the square or quadratic formula) to find the solutions for x. Please refer to the Factoring Polynomials article if you need a reminder of this.
After doing polynomial division and factoring of the polynomial function , we get the result .
Based on this, the zeros or x-intercepts are:
, and
If a zero appears as part of the solution twice (it is repeated), then the curve of the function will touch the x-axis at that value of x, and then bounce off the x-axis changing its direction.
2. Find the y-intercept: Substitute x = 0 in the original polynomial function. The result will be the y-coordinate where the curve crosses the y-axis.
The point where the curve of the function crosses the y-axis is (0, -12)
3. End Behavior: The curves of polynomials that have a degree of 2 or more are continuous and smooth lines that can have maximum or minimum points where they change direction in the middle section of the curve, and on either end of the curve, they tend to go towards positive or negative infinity.
How do you determine the end behavior of a function?
Leading Coefficient Test: As mentioned before, the leading term of a polynomial is the term with the highest exponent. You will need to look at whether its exponent is even or odd and the sign of its coefficient to help you determine the end behaviour of the curve.
Odd function (i.e. )
a) Positive leading coefficient: In this case, the function will point downwards on the left and point up on the right end of the curve.
b) Negative leading coefficient: In this case, the function will point up on the left and point down on the right end of the curve.
Even function (i.e. )
a) Positive leading coefficient: In this case, the function will point upwards on both ends of the curve.
b) Negative leading coefficient: In this case, the function will point downwards on both ends of the curve.
To sum up the end behavior of the polynomial graph we have the table below:
Odd function | Even function | |||||||
Sign of the Leading Coefficient | Positive | Negative | Positive | Negative | ||||
End Behavior | Left | Right | Left | Right | Left | Right | Left | Right |
↓ | ↑ | ↑ | ↓ | ↑ | ↑ | ↓ | ↓ |
The leading term of the polynomial function is , which means that it is an odd function with a positive leading coefficient. Therefore the end behaviour of the curve will be like this:
Left | Right |
↓ | ↑ |
4. Sketch the curve of the function.
Another way of graphing polynomials in which you do not need to have the polynomial in factored form is as follows:
Make a table of values using direct substitution of a few values of x.
Plot the points on the coordinate plane to determine the middle section of the graph.
Connect the points with a smooth and continuous curve.
Use the leading coefficient test to determine the end behaviour of the graph.
Polynomials are expressions involving multiple terms that contain a variable raised to a series of positive whole-number exponents. Each term may also be multiplied by coefficients.
Evaluating the value of a polynomial consists of finding the solution of the polynomial for a specific value of x.
You can evaluate a polynomial using two methods: direct substitution or synthetic substitution.
According to their degree, polynomial functions will have different types of graphs. The curves of polynomials that have a degree of 2 or more are continuous and smooth lines that can have maximum or minimum points where they change direction in the middle section of the curve, and on either end of the curve, they tend to go towards positive or negative infinity. The degree of a polynomial matches the number of direction changes in their graph and the number of zeros or x-intercepts.
To graph polynomial functions follow these steps:
Alternatively, you can make a table of values using direct substitution of a few values of x, and plot the points on the coordinate plane to determine the middle section of the graph. Then, you can connect the points with a smooth and continuous curve, and use the leading coefficient test to determine the end behaviour of the graph. In this case, you do not need to have the polynomial in factored form.
What is a polynomial?
Polynomials are expressions involving multiple terms that contain a variable raised to a series of positive whole-number exponents. Each term may also be multiplied by coefficients.
What is the leading term of the polynomial?
The leading term of a polynomial is the term with the highest exponent.
What is the leading coefficient of a polynomial?
The leading coefficient is the coefficient of the leading term of the polynomial.
What is the degree of a polynomial?
The degree of a polynomial is the highest power or exponent present in the polynomial.
What is evaluating the value of a polynomial?
Evaluating the value of a polynomial consists of finding the solution of the polynomial for a specific value of x.
What methods can be used to evaluate a polynomial?
Direct substitution and synthetic substitution.
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