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Solving Quadratic Equations by Completing the Square
Graphing Logarithmic Functions
Division Property of Exponents
Adding and Subtracting Rational Expressions With Like Denominators
Rationalizing the Denominator
Multiplying Special Polynomials
Functions
Solving Linear Systems of Equations by Elimination
Solving Systems of Equation by Substitution and Elimination
Polynomial Equations
Solving Linear Systems of Equations by Graphing
Quadratic Functions
Solving Proportions
Parallel and Perpendicular Lines
Simplifying Square Roots
Simplifying Fractions
Adding and Subtracting Fractions
Adding and Subtracting Fractions
Solving Linear Equations
Inequalities in one Variable
Recognizing Polynomial Equations from their Graphs
Scientific Notation
Factoring a Sum or Difference of Two Cubes
Solving Nonlinear Equations by Substitution
Solving Systems of Linear Inequalities
Arithmetics with Decimals
Finding the Equation of an Inverse Function
Plotting Points in the Coordinate Plane
The Product of the Roots of a Quadratic
Powers
Solving Quadratic Equations by Completing the Square

Polynomial Equations

Polynomial Equations in Disguise

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

y = c0 + c1·x + c2·x2 + ... + cn·xn

where the powers of x must be positive integers and the letters c0, c1, … , 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 – 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 · (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 · (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 – 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 · (0 - 3)2 +1. 

4 = a · 9 +1. 

3 = a · 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 – 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 · (x -1) · (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 · (0 -1) · (0 - 4) 

-2 = a · 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 “-” 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.

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