Pre-AP Pre-Calculus Chapter 2, Section 4

Real Zeros of Polynomial Functions

2013 - 2014

Long Division

• Factoring polynomials reveals its zeros.

• Polynomial division gives another way to factor polynomials.

32 3587

3𝑥 + 2 3𝑥3 + 5𝑥2 + 8𝑥 + 7

Something to remember

• Each term of the polynomial must be represented.

– Example:

Use long division to find the quotient and remainder when 2𝑥4 − 𝑥3 − 2 is divided by

2𝑥2 + 𝑥 + 1

𝑥 + 4 3𝑥2 + 7𝑥 − 20

Remainder and Factor Theorem

• Used when the divisor is in the form 𝑦 = 𝑥 − 𝑘

• Remember: The factor is (𝑥 − 𝑘), but the zero of the function is 𝑘

• 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: 𝑓 𝑘 = 𝑟

Find the remainder when

𝑓 𝑥 = 3𝑥2 + 7𝑥 − 20

is divided by 𝑥 − 2.

Apply Remainder Theorem

Find the remainder when 𝑓 𝑥 = 3𝑥2 + 7𝑥 − 20 is divided by 𝑥 + 1.

Apply Remainder Theorem

Find the remainder when 𝑓 𝑥 = 3𝑥2 + 7𝑥 − 20 is divided by 𝑥 + 4.

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 2𝑥3 − 3𝑥2 − 5𝑥 − 12 by 𝑥 − 3

Practice Synthetic Division

• Divide 𝑥3 − 5𝑥2 + 3𝑥 − 2 by 𝑥 + 1

Practice Synthetic Division

• Divide 2𝑥4 − 5𝑥3 + 7𝑥2 − 3𝑥 + 1 by 𝑥 − 3

Rational Zeros Theorem

• Zeros of polynomial functions are either rational zeros or irrational zeros.

• 𝑓 𝑥 = 4𝑥2 − 9 = (2𝑥 + 3)(2𝑥 − 3)

• 𝑓 𝑥 = 𝑥2 − 2 = (𝑥 + 2)(𝑥 − 2)

Rational Zeros Theorem

• Suppose f is a polynomial function of degree 𝑛 ≥ 0 of the form

𝑓 𝑥 = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1+. . . + 𝑎0

Where every coefficient is an integer, 𝑎0 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 𝑝

𝑞.

• 𝑓 𝑥 = 2𝑥4 − 7𝑥3 − 8𝑥2 + 14𝑥 + 8 • What are the factors of p (8)?

• What are the factors of q (2)?

• Now list all possible real zeros of the function 𝑝

𝑞=

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

zero.

Example continued

• Once you determine if 𝑝

𝑞 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.

𝑓 𝑥 = 6𝑥3 − 5𝑥 − 1

Using Rational Zero Theorem

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

𝑓 𝑥 = 2𝑥3 − 𝑥2 − 9𝑥 + 9

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 𝑓(𝑥) is divided by x – k by using synthetic division

– If 𝑘 ≥ 0 and every number in the last line is a nonnegative (0 or positive) then k is an upper bound

– If 𝑘 ≤ 0 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 𝑓 𝑥 = 2𝑥4 − 7𝑥3 − 8𝑥2 + 14𝑥 + 8

are in the interval [-2, 5].

Find all the real zeros of 𝑓 𝑥 = 2𝑥4 − 7𝑥3 − 8𝑥2 + 14𝑥 + 8

𝑓 𝑥 = 10𝑥5 − 3𝑥2 + 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 5, 9, 15, 17, 23, 25, 27, 37, 43, 49, 57, 63, 67

• 13 Total problems

• Gray Book: pages 205 - 207