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# Polynomial Equations

## Polynomial Equations in Disguise

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

y = c0 + c1Â·x + c2Â·x2 + ... + cnÂ·xn

where the powers of x must be positive integers and the letters c0, c1, â€¦ , cn 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.