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 Depdendent Variable

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 Dependent Variable

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# Polynomial Equations

Solve the quadratic inequality:  12Â· x2 - 22 Â· x 4.

Solution

The graphs of the quadratic (  y =12 Â· x2 - 22Â· x) and the horizontal line ( y = 4) are shown together in the graph below. As we are looking for the x-values that make the height of the quadratic less than the height of the horizontal line, our answer will consist of all xvalues that lie between the two intersection points (including the intersection points themselves as the â€œâ€ sign is used, rather than the â€œ<â€ sign.)

To locate the x-coordinates of the points where the quadratic and the horizontal line meet, we must manipulate the given inequality to make the right hand side of the inequality equal to zero. To do this we can subtract 4 from both sides.

12 Â· x2 - 22 Â· x - 4 0. (Subtract 4 from both sides)

To find the x-coordinates of the intersection points, you need to solve the quadratic equation:

12 Â· x2 - 22 Â· x - 4 = 0.

So, the solution to the quadratic inequality consists of the intervals of x-values:

## Polynomial Equations

A polynomial equation is an equation of the form:

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

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

The largest power of x in the polynomial is called the degree of the polynomial.

Each of the different types of polynomial functions has a distinctively shaped graph. Some of the polynomials (and shapes) that you are already familiar with include:

Degree 0 polynomials (i.e. constant functions) have graphs that are horizontal lines.

Degree 1 polynomials (i.e. linear functions) have graphs that are straight lines.

Degree 2 polynomials (i.e. quadratic functions) have graphs that are parabolas.

Degree 3 polynomials (called cubics) have graphs that look like John Travolta from the cover of the Saturday Night Fever movie soundtrack (see Figure 1).

Figure 1: The graph of a polynomial function of degree 3.