Polynomials

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How to identify the degree of a polynomial

By end behavior/+ or - leading coefficient/even or odd degree

By turning points: Depending on how many turning oppoints a polynomial has, the degree is always one number higher.

Quintic function(four turning points)=Degree of 5

Quartic function(three turning points)=Degree of 4

Cubic function(two turning points)=Degree of 3

Quadratic function(one turning point)=Degree of 2

Linear function(no turning points)=Degree of 1

Odd

RISES to the left and FALLS to the right

FALLS to the left and RISES to the right

Even

- leading coefficient

FALLS to both left and right

+ leading coefficient

RISES to both left and right

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

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

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

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

How to solve a polynomial

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.

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 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 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.

How to identify the number of real or complex zeros

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

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.

Definitions

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

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

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.

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

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

Main topic

Albert Kalinin