A polynomial function is a function that can be written as:

f(x)=axⁿ+bxⁿˉ¹+cxⁿˉ²…gx+h

Where :

a, b, c, …, g, and h are real numbers

n is a positive integer.

POLYNOMIALS NOT POLYNOMIALS

f(x)=½x²+2x-3

Why? Correct form

½ , 2, and 3 are real numbers

2 and 1 are both positive integers.

F(x)= ίxˉ²

Why not? ί is not a real number

-2 isn’t a positive number

Standard form:

f(x)=axⁿ+bxⁿˉ¹+cxⁿˉ²…gx+h

Everything in the equation is multiplied together.

Factored form:

f(x)=(x-r1)(x-r2)(x-r3)…

Degree: the highest power of “x” in standard form.

Ex. : f(x)=½x²+2x-3 Degree = 2

Leading coefficient: coefficient of the first term if the terms are in descending order.

f(x)=½x²+2x-3 Leading Coefficient = ½

The factors of a polynomial are the quantities of the function when in factored form

Ex: f(x)=(x-5)(x+7)(x-2)

There are three factors in this polynomial. They are x-5, x+7, and x-2.

The zeros of a function are the numbers which can be inserted into “x” so that the function equals zero.

Example:

f(x)=(x-4)(x+7) f(x)=(x-4)(x+7)

f(4)=(4-4)(4+7) f(-7)=(-7-4)(-7+7)

f(4)=(0)(11) f(-7)=(-11)(0)

f(4)=0 f(-7)=0

The zeros of this function are 4 and -7.

When graphed, the zeros are the x-intercepts.

The number of zeros in a function are the same as the intercepts. Therefore, an equation with three different factors, such as

f(x)=(x-2)(x+3)(x-7), would have three x-intercepts at (2,0), (-3,0), and (7,0).

When two of the factors in an function are the same (ex.: f(x)=(x-2)(x+3)(x-2)), then the graph will, in this case, have two x-intercepts instead of three, with the curved line sitting on the x-axis instead of crossing it and intersecting in three places.

The graph of a function cannot intersect the x axis any more time than the number of factors it ha.

When the factors of a function with just two factors are the same (ex.: f(x)=(x+3)(x+3)), the function has a “multiplicity of two.”