PRE-AP PRE-CALCULUS

CHAPTER 2, SECTION 4

Real Zeros of Polynomial Functions2013 - 2014

LONG DIVISION

Factoring polynomials reveals its zeros.

Polynomial division gives another way to factor polynomials.

3 𝑥+23𝑥3+5 𝑥2+8 𝑥+7

SOMETHING TO REMEMBER Each term of the polynomial must be represented. Example:

USE LONG DIVISION TO FIND THE QUOTIENT AND REMAINDER WHEN IS DIVIDED BY

𝑥+4 3𝑥2+7 𝑥−20

REMAINDER AND FACTOR THEOREM Used when the divisor is in the form

Remember: The factor is , but the zero of the function is

If you use the Remainder/Factor Theorem, and you get a number, that number is a remainder.

If you use the Remainder/Factor Theorem, and you get 0, then the value of k is a zero of the function.

APPLY REMAINDER THEOREM

Theorem equation: Find the remainder when

is divided by .

APPLY REMAINDER THEOREM

Find the remainder when is divided by .

APPLY REMAINDER THEOREM

Find the remainder when is divided by .

THEOREM FACTOR THEOREM A polynomial function f(x) has a factor of x – k if and only if f(k) = 0.

FACTORING VS. DIVISION Factoring is easier to use when polynomial degrees are 3 or less.

When polynomial degrees are higher than 3, division would be the way to go.

SYNTHETIC DIVISION

Used when the divisor is the linear function x – k

http://www.youtube.com/watch?v=bZoMz1Cy1T4

PRACTICE SYNTHETIC DIVISION Divide by

PRACTICE SYNTHETIC DIVISION Divide by

PRACTICE SYNTHETIC DIVISION Divide by

RATIONAL ZEROS THEOREM Zeros of polynomial functions are either rational zeros or irrational zeros.

RATIONAL ZEROS THEOREM Suppose f is a polynomial function of degree of the form

Where every coefficient is an integer, does no equal zero, and you cannot factor out a constant, then

p is an integer factor of the constant coefficient

q is an integer factor of the leading coefficient

EXAMPLE OF RATIONAL ZEROS THEOREM The possible rational zeros would be .

What are the factors of p (8)?

What are the factors of q (2)?

Now list all possible real zeros of the function

Plug the values in the calculator and see if they are in fact a real zero.

EXAMPLE CONTINUED Once you determine if is a real zero, use synthetic division to find the other factor of the polynomial.

USING RATIONAL ZERO THEOREM

Find all possible zeros of the given function, then determine which ones (if any) are actual zeros.

USING RATIONAL ZERO THEOREM

Find all possible zeros of the given function, then determine which ones (if any) are actual zeros.

UPPER AND LOWER BOUNDS You can find an interval that all the real zeros occur in a function – they are called upper and lower bounds.

If you find an upper bound for real zeros, that means the graph will NOT pass through the x-axis at any number higher than the upper bound.

If you find a lower bound for real zeros, that means the graph will NOT pass through the x-axis at any number lower than the lower bound.

FINDING UPPER AND LOWER BOUNDS The polynomial must have a positive leading coefficient, and the exponent must be ≥ 1

Suppose is divided by x – k by using synthetic division If and every number in the last line is a nonnegative (0 or positive)

then k is an upper bound If and the numbers in the last line are alternately nonnegative and

a positive, the k is a lower bound.

***Just be k is a bound, does NOT mean it is a zero of the function!

ESTABLISHING BOUNDS FOR REAL ZEROS Prove that all of the real zeros of

are in the interval [-2, 5].

FIND ALL THE REAL ZEROS OF

𝑓 (𝑥 )=10𝑥5−3 𝑥2+𝑥−6 Prove the zeros occur in the interval [0, 1].

Find all the possible zeros of the function.

Determine with ones are the actual zeros.

CH. 2.4 HOMEWORK

Pg. 223 – 226: #’s 4, 8, 15, 18, 22, 25, 26, 27, 38, 43, 49, 57, 64, 67

14 Total problems

Gray Book: pages 205 - 207