Polynomials

Albert Kalinin

Main topic

Definitions

X-intercepts/zeros: The x intercepts are found when the polynomial on a graph touches the x-axis. It can also be found by using the quadratic formula on the polynomial equation.

Polynomials: An expression consisting of the sum of two or more terms each of which is the product of a constant and a variable raised to an integral power

End behavior: If a line is finite and has no end, it will show end behavior either being negative, sloping towards negative infinity, or positive, sloping towards positive infinity.

Leading coefficient: This is the number in the polynomial equation that had the highest degree

Turning points: The number of turning points in a polynomial tells us how high the degree of it is.

How to identify the number of real or complex zeros

To determine the number of zeros on a graph you look at the amount of times the polynomial intersects or crosses the x-axis

If there is no graph to look at then you can determine the number of zeros by factoring the polynimial equation.

The number of complex zeros can be determined by using synthetic division and then by factoring

How to solve a polynomial

If not given any zeros

If you are not given any zeros, you must complete the rational zero test.

The rational zero test requires you to use the remainder of the polynomial equation and the leading coefficient to find zeros.

If given a real zero

If you are given a zero, then synthetic division must be preformed on the polynomial equation.

One you get your answer from the division, you then factor is and find your zeros, including your original zero

If given a complex zero

If given a complex zero, you must use synthetic division on the polynomial equation.

Once this is done, factor the answer from the division and use the property of imaginary numbers if needed.

If given two real zeros

If you are given two real zeros, then you must find out what degree the zeros are before you can begin.

In this, you must preform synthetic division twice . Once and then another time on the answer of the first division with the second zero.

After this is preformed you must factor and find any possible remaining zeros.

How to identify the degree of a polynomial

By zeros: The degree of the polynomial can be determined by what happens to the zeros along the x-axis.

If the polynomial crosses the x-axis, it is determined as having no degree.

If the polynomial bounces off of the x-axis, the zero is a degree of 2.

If the polynomial "squiggles" across the x-axis, the zero has a degree of 3.

Even

+ leading coefficient

RISES to both left and right

- leading coefficient

FALLS to both left and right

Odd