How to Graph a Polynomial

Have you been taught how to graph a polynomial, but wasn't quite sure what the teacher was talking about??

Check out this instructable to learn it again

Pens and pencils

Ruler
Paper
Calculator

First up, look at your polynomial. One of the numbers, will not have an x attached to it. This is your constant.

Find the factors of the constant. Factors are all the numbers the constant can be divided by.

After you have found factors of the constant, you now need to use the factor theorem to find a factor of the polynomial.

The factor theorem says that for P(x), if P(k) = 0 then x - k is a factor of P(x).

Substitute each of the factors of your constant into P(x), one at a time. If it equals zero, you've found your factor.

See image for more information.

After you have found your factor, in my case x - 1, divide the polynomial by the factor.

Write it out like a long division problem you might have done in elementary, middle or primary school. (Depending on where you went to school).

Divide the first term by x. Put the number you are left with above the line. Then, times the number above the line by x. You should have the same number you started with. Write this below the original number. Then times the number above the line by the constant, in my case, negative 1. Write this next to the number written below the original number.

Take these two numbers away from the first two number sitting under the line. The result is what you use to repeat the process with again.

Keep going until you are left with one number. If you have done this right, it should be equal zero (0)

From the images, you should be able to see how I repeated what I did in steps 1 to 3, until I was left with one x squared, but no x cubed or more.

This is pretty simple. Use either the quadratic formula, or distributive property to do this.

I found two number that equalled the co-efficient of x when added together, and equalled the constant when multiplied.

See image if this is confusing.

By now, all your terms should look like this (x+k) or (x-k) where k is just a regular old number.

Now you want to put all your terms in the one line, so that you have P(x) again, just written differently, factored to be exact.

Now we start to find information we will put on the graph.

The first thing we find are all the numbers where the graph crosses the x-intercept, or the horizontal axis.

x-Intercepts are whatever makes each bracket equal zero (0) because if this number were substituted into the polynomial, the result would be 0.

0 times anything is 0

To find the y-intercept we have to substitute in x=0. So wherever we saw an x, we have to put a zero. Then just solve the equation.

Drawing a blank cartesian plane is pretty simple. Make sure you use a ruler, to ensure lines are straight.

Then put in the x- and y-intercepts you have found already.

This step is important, but easily overlooked. It ensure that the line goes on the correct side of the x or horizontal axis.

Testing points is a simple as substituting different values of x into the original, or factorised, polynomial. Make sure you substitute in one value between each point on the graph. You might have to stick with this step for a while to get them all done. That's okay.

Now add it all together.

Join the dots between the intercepts, making sure that the line goes on the correct side of the horizontal or x-axis.

And you're done!! Congratulations. You have graphed a polynomial. Phew!!