Zeros of p(x)

Reynaldo B. Pantino, T2

Finding the Zeros of a Polynomial Function

Objectives

1.) To determine the zeros of polynomial functions of degree greater than 2 by;

a.) factor theoremb.) factoringc.) synthetic divisiond.) depressed equations

2.)To determine the zeros of polynomial functions of degree n greater than 2 expressed as a product of linear factors.

Recapitulations

What is remainder theorem?

What is synthetic division?

What is factoring?

What is zero of a function?

Discussions

UNLOCKING OF DIFFICULTIES

The zero of a polynomial function P(x) is the value of the variable x, which makes polynomial function equal to zero or P(x) = 0.

Discussions

UNLOCKING OF DIFFICULTIES

The fundamental Theorem of Algebra states that “Every rational polynomial function

P(x) = 0 of degree n has exactly n zeros”.

Discussions

UNLOCKING OF DIFFICULTIES

When a polynomial is expressed as a product of linear factors, it is easy to find the zeros of the related function considering the principle of zero products.

Discussions

UNLOCKING OF DIFFICULTIES

The principle of zero product state that, for all real numbers a and b, ab = 0 if and only if a = 0 or b = 0, or both.

Discussions

UNLOCKING OF DIFFICULTIES

The degree of a polynomial function corresponds to the number of zeros of the polynomial.

Discussions

UNLOCKING OF DIFFICULTIES

A depressed equation of P is an equation which has a degree less that of P.

Discussions

Illustrative Example 1Find the zeros of P(x) = (x – 3)(x + 2)(x – 1)(x + 1). Solution: (Use the principle of zero

products)P(x) = 0; that is

x - 3 = 0 x + 2 = 0 x - 1 = 0x + 1 = 0

x = 3 x = -2 x = 1 x = -1

Discussions

Illustrative Example 2Find the zeros of P(x) = (x + 1)(x + 1)(x +1)(x – 2)Solution: (By zero product principle)we have, P(x) = 0 the zeros are -1

and 2.The factor (x + 1) occurs 3 times. In this

case, the zero -1 has a multiplicity of 3.

Discussions

Illustrative Example 3Find the zeros of P(x) = (x + 2)3(x2

– 9).Solution: (By factoring)we have, P(x) = (x +2)(x+2)(x+2)(x –

3)(x + 3).The zeros are;

-2, 3, -3,

where -2 has a multiplicity of 3.

Discussions

Illustrative Example 4 Function Zeros No.

of Zero

sP(x) = x – 4

P(x) = x2 + 8x + 15P(x) = x3 -2x2 – 4x + 8

P(x) = x4 – 2x2 + 1

4 1

32, -2, 2

-3, -5 2

1,1,-1,-1 4

Discussions

Illustrative Example 4 Solve for the zeros of P(x) = x3 + 8x2 + 19x + 12, given that one

zero is -1.Solution: By factor theorem, x + 1 is a factor

of x3 + 8x2 + 19x + 12.

Then; P(x) = x3 + 8x2 + 19x + 12= (x+1)● Q(x).

Discussions

Illustrative Example 4 (Continuation of solution)

To determine Q(x), divide x3 + 8x2 + 19x + 12 by

(x + 1). By synthetic division;-1-1 11 88 1919 1212

11-1-177

-7-71212

-12-1200

Discussions

The equation x2 + 7x + 12 is a depressed equation of P(x). To find the remaining zeros use this depressed equation.By factoring we have;

x2 + 7x + 12 = 0(x +3)(x + 4) = 0 x = -3 and x = -4

Therefore; the three zeros are -1, -3, and -4.

Observe that a polynomial function of degree 3 has

exactly three zeros.

Illustrative Example 4 (Continuation of solution)

Exercises

1. Solve for the other zeros ofP(x) = x4 – x3 – 11x2 + 9x + 18, given that one zero is -3.

2. Solve for the other zeros ofP(x) = x3 – 2x2 – 3x + 10, given that – 2 is a zero.

Activity Numbers

Which of the numbers -3, -2, -1, 0, 1, 2, 3 are zeros of the following polynomials?

1.) f(x) = x3 + x2 + x + 12.) g(x) = x3 – 4x2 + x + 63.) h(x) = x3 – 7x + 6 4.) f(x) = 3x3 + 8x2 – 2x + 3 5.) g(x) = x3 + 3x2 – x – 3

Activity Factors

Which of the binomials (x – 1), (x + 1), (x – 4), (x + 3) are factors of the given polynomials.

1.) x3 + x2 - 7x + 52.) 2x3 + 5x2 + 4x + 13.) 3x3 – 12x2 + 2x – 8 4.) 4x4 - x3 + 2x2 + x – 3 5.) 4x4 + 5x3 - 14x2 – 4x + 3

Activity Zeros

Find the remaining zeros of the polynomial function, real or imaginary, given one of its zeros.1.) P(x) = x3 + 5x2 - 2x – 24 x = 22.) P(x) = x3 - x2 - 7x + 3 x = 33.) P(x) = x3 – 8x2 + 20x – 16 x = 24.) P(x) = x3 + 5x2 - 9x – 45 x = -55.) P(x) = x3 + 3x2 + 3x + 1 x = -1

Assignments

On page 103, answers numbers 6, 12, 18,19, & 20.Ref. Advanced Algebra, Trigonometry & Statistics

What is rational Zero Theorm? Pp. 105