Chapter 4 – Polynomial and Rational Functions

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Presentation transcript:

Chapter 4 – Polynomial and Rational Functions Section 1 – Polynomial Functions

Polynomial Functions A polynomial function is any function like 4x3 + 2x2 + 4. The only requirements are that the leading term must be first and has the degree of the polynomial attached to it (degree 3). To be considered a polynomial, all exponents must be nonnegative integers.

Polynomial Functions ZEROS of a polynomial are values of x for which f(x) is zero, in other words, these are x-intercepts. Every polynomial can be expressed as the product of a constant and/or a number of linear factors equal to the degree of the polynomial.

Fundamental Theorem of Algebra The FTA states “Every polynomial equation with degree n greater than zero has exactly n roots.” The most zeros a function can have is the same as its degree. (Degree 2 can have 2 zeros). Ex. If -2 is a zero, (x+2)is a factor If 5 is a zero, (x – 5) is a factor.

Try it out. EX 1: Consider the polynomial function f(x)= x3 - 6x2 + 10x – 8 A) State the degree and leading coefficient of the polynomial B) Determine whether 4 is a zero of f(x)

More About Zeros An odd-degree function has to have at least ONE REAL zero. (EX: f(x)= x3 + 6) An even-degree function can have NO REAL ZEROS or it will have an even number of real zeros. (EX: f(x)= x2 + 5 and x2 – 6) **unless a vertex is the zero! Then only one! But it is a “double” (multiplicity 2)**

If We Know the Roots The roots of a polynomial function are the zeros of the function. These two terms can be used interchangeably. Well, if we know the roots, then we can write the equation. EX: If 3 is a root of a function, then we know the function has to have the piece (x – 3).

MORE!!! EX 2: Write a polynomial equation of least degree with roots 2, 4i, and -4i. A) Does the equation have an odd or even degree? How many times does the graph of the function cross the x-axis?

One Last One EX 3: Factor the polynomial f(x)= 9x4 - 35x2 – 4 to find the roots and then we’ll graph the function. (hint: look at the relationship between the coefficients to help factor. Imaginary roots are NOT x-intercepts)

Assignment Chapter 4, Section 1 pgs 209-211 #6-12E,16-46E,52,56