## Polynomials Introduction

A polynomial is any combination of power functions where n is equal to a positive integer or zero.

$y=a_{0}+a_{1}x+a_{2}+x^{2}+.......+a_{n}x^{n}$

Polynomial Types

• A degree one polynomial is where there is x to the power of one plus any real number, a linear function
• A degree two polynomial is the same a quadratic function
• A degree three polynomial is called a cubic.

### Turning and Inflection Points

When a function has an even amount of turning points at the same point, we have a point of inflection in our graph.

A point of inflection is characterised by a gradient which approaches zero and is 0 at its point, just like a turning point. However unlike the turning point it then continues in the same direction instead of reversing

For the blue cubic function there is a point of inflection at (-2,0), since both the turning points are located at the same point. Note how the function flattens out and then continues in the same direction. A cubic function is also able to have two turning points at different co-ordinates instead of a point of inflection.

For the red quadratic function, the turning point we can see at (2,0). Note how the function flattens out and then reverses direction.

### Turning Point Form

Sometimes a polynomial equation may appear in a form similar to the turning point form of a quadratic, but its not possible to display all polynomials in this form unlike a quadratic:

$y=a(x+h)^{n}+k$

This form lets us know where the turning point or the point of inflection is straight away.

If n is a positive odd integer greater or equal to 3 then we know that it will be a point of inflection at (-h,k). In this form all the turning points are located in the one spot.

If n is a positive even integer greater or equal to 2 then we know that it will be a turning point at (-h,k). This is the same form as our turning point form for quadratics.

To draw the rest of the graph we go through the same process as the quadratic of setting x to zero to calculate the y intercept and then y to zero to calculate the x intercept.

### Example

For example if we were required to sketch:

$y=2(x+1)^{3}+2$

Step 1: Locate the point of Inflection

As the function is in turning point form and to the power of an odd number the x co-ordinate of the point of inflection is easy to find. In this case it is (-1,2).

Step 2: Find the y intercept, by setting x to zero

$y=2(0+1)^{3}+2\newline =4$

Step 3: Find the x intercept by setting y to zero

$y=2(x+1)^{3}+2=0\newline 2(x+1)^{3}=-2\newline x+1=\sqrt[3]{\frac{-2}{2}}\newline x=\sqrt[3]{-1}-1=-2$

Step 4: Connect the dots in a polynomial shape like so:

Receiving a polynomial in turning point form is easiest to sketch as it will always be either the S shape of the cubic above when is an odd number or the U shape when n is an even number. However, since not all polynomials can be put in this form, they are not all this easy to sketch.