# Polynomial Equations

## Polynomial Equations in Disguise

The standard format (or **standard form**) for the formula of a polynomial
equation is:

y = c_{0} + c_{1}Â·x + c_{2}Â·x^{2} + ... + c_{n}Â·x^{n}

where the powers of x must be positive integers and the letters c_{0}, c_{1}, â€¦ , c_{n} represent
numbers.

The formulas of polynomial equations sometimes come expressed in other formats, such
as factored form or vertex form. The specific format that the formula of a polynomial
equation is expressed in does not matter so much â€“ you can always convert the formula to
standard form by foiling to check that the formula really is the formula of a polynomial
equation.

**Example **

Figure 2 shows the graph of a quadratic equation.

(a) Find a formula for the quadratic equation expressed in vertex form.

(b) Find a formula for the quadratic equation expressed in standard form.

(c) Is the equation that you have found a formula for a polynomial equation or not?

Figure 1: Find the formula of this quadratic equation.

**Solution**

(a) The vertex form of a quadratic equation looks like:

y = a Â· (x - h)^{2} + k,

where the letter h is the x-coordinate of the vertex and the letter k is the ycoordinate
of the vertex. Figure 1 shows that the x-coordinate of the vertex is
equal to 3 and that the y-coordinate of the vertex is equal to 1. This means that
the vertex form of this quadratic will be:

y = a Â· (x - 3)^{2} +1.

All that remains is to find the numerical value of the constant a. To do this, you
can use the x- and y-coordinates of any other point (i.e. other than the vertex) that
lies on the quadratic â€“ for example the point (0, 4) shown in Figure 2. To work
out the value of a we will plug x = 0 and y = 4 into the vertex form and then solve
for a.

4 = a Â· (0 - 3)^{2} +1.

4 = a Â· 9 +1.

3 = a Â· 9.

So, the equation for the quadratic equation shown in Figure 1 (expressed in vertex
form) is:

To convert this equation from vertex form to standard form, you can expand by
FOILing and then collect like terms.

(Expand the (x â€“ 3)2 by FOILing)

(Multiply through by one third)

(Combine the like terms)

So, the equation for the quadratic equation shown in Figure 2 (expressed in
standard form) is:

(c) This is a polynomial equation because the formula consists of powers of x added
together. All of the powers that appear are positive integers.

**Example **

Figure2 (see below) shows the graph of a quadratic equation.

(a) Find a formula for the quadratic equation expressed in factored form.

(b) Find a formula for the quadratic equation expressed in standard form.

(c) Is the equation that you have found a formula for a polynomial equation or not?

Figure 2: Find the formula of this quadratic equation.

**Solution**

(a) The x-intercepts of the quadratic shown in Figure 2 are located at x = 1 and x = 4.
This means that the factored form of the quadratic equation must look something
like this:

y = a Â· (x -1) Â· (x - 4).

The factored form must have a factor of (x - 1) to ensure that when you plug in x
= 1 the value of y will be equal to zero. The factored form must also have a factor
of (x - 4) to ensure that when you plug in x = 4 the value of y will be equal to
zero.

To determine the numerical value of a you can plug in the x- and y-coordinates of
any other point on the quadratic graph (i.e. any point other than one of the xintercepts)
and solve for a. Figure 2 shows that the point (0, -2) lies on the graph,
so you can plug in x = 0 and y = -2 into the factored form. Doing this:

-2 = a Â· (0 -1) Â· (0 - 4)

-2 = a Â· 4

So, the equation of the quadratic equation from Figure 2 (written in factored form)
is:

(b) To convert this equation to standard form, you can expand by FOILing and then
simplify (if necessary). Doing this:

(Expand by FOILing)

(Multiply through by
- beware of â€œ-â€ signs)

(c) This is a polynomial equation because the formula consists of powers of x added
together. All of the powers that appear are positive integers.