POLYNOMIALS AND THEIR ZEROS - Austin Community College ...

POLYNOMIALS AND THEIR ZEROS

A polynomial of degree n may always be written in

a standard form:

f (x) an x n an1x n1 ... a1x a0

where an is the leading coefficient and (0, a0 ) is the y-intercept, or

a complete factored form: f (x) an (x xn )(x xn1) ... (x x1)

where an is the leading coefficient and the numbers x1, ..., xn are zeros of the polynomial

f, which means that:

f (xn) 0, f (xn1) 0,...,f (x1) 0

In general, the zeros may be complex numbers.

This is at the heart of The Fundamental Theorem of Algebra whose consequence is that a polynomial of degree n has exactly n complex zeros, where complex numbers include real numbers.

Note: If a number z is a real zero of a function f, then a point (z, 0) is an x-intercept of the graph of f.

The non-real zeros of a function f will not be visible on a xy-graph of the function.

Examples:

Standard f (x) 3x2 3x 6 h(x) 2x4 4x3 4x 2 g(x) 2x4 4x3 4x2 4x 2

Form

Factored f (x) 3(x 1)(x 2) h(x) 2(x 1)(x 1)3

Form

g(x) 2(x 1)2 (x i)(x i)

Zeros x1 1, x2 2

x1 1, x2 1, x3 1, x4 1

x1 1, x2 1, x3 i, x4 i

FINDING ZEROS OF POLYNOMIALS

If f (x) an(x xn)(x xn1) ...(x x1) then the zeros are shown explicitly ( x1,...,xn ) but if f is not given in a complete factored form then depending on the degree different techniques apply.

Examples

For a polynomial of degree 2, a quadratic function, we can always use the Quadratic Formula to find the zeros. In some cases, factoring is possible instead.

1. Let f (x) 3x2 3x 6 . Find the zeros of f, i.e. solve f(x) = 0

Factoring

Quadratic Formula

3x2 3x 6 0 3(x2 x 2) 0 3(x 2)(x 1) 0 x 2 0 x 1 0 x 2 x 1

a 3 b 3 c 6 OR x b b2 4ac

2a 3 (3)2 4(3)(6) x

2(3)

x 3 81 3 9

6

6

x1

39 6

2

x2

3 6

9

1

2. Let f (x) x2 8 . Find the zeros of f, i.e. solve f(x) = 0.

Factoring x2 8 0

Quadratic Formula a 1 b 0 c 8

(x i 8)(x i 8) 0

x i 8 i2 2

or

OR

x i2 2

x b b2 4ac 2a

0 x

(0)2 4(1)(8)

32 4

2

2(1)

2

2

x1 i2 2 x2 i2 2

FINDING ZEROS OF POLYNOMIALS

For a polynomial of degree n > 2 we can try factoring techniques. If they do not apply easily or at all, there are theorems that help in narrowing down the candidates for zeros. To check if a particular number, x1, indeed is a zero of a polynomial we can divide the polynomial by the factor (x ? x1). If the remainder is equal to zero than we can rewrite the polynomial in a factored form as (x x1) f1(x) where f1(x) is a polynomial of degree

n 1 . This process can be continued until all zeros are found.

Factoring f (x) 2x4 4x3 4x 2

f (x) 2(x 1)(x 1)3 0

x1 1 x2 1 x3 1 x4 1

Division by linear factors of the form x - c Is x = 1 a zero of f? Use synthetic division to check if (x ? 1) divides f without a remainder:

1 -2 -4 0 4 2 -2 -6 -6 -2

____________________ -2 -6 -6 -2 0 Since the remainder = 0 then the polynomial f can be rewritten as

f (x) (x 1) f1(x) (x 1)(2x3 6x2 6x 2) Is x = 1 a zero of f1? 1 -2 -6 -6 -2

-2 -8 -14 ____________________ -2 -8 -14 -16 Since the remainder is not 0 then the polynomial f has only one zero x = 1.

Is x = -1 a zero of f ? -1 -2 -6 -6 -2

2 42 ____________________ -2 -4 -2 0 Since the remainder = 0 then the polynomial f can be rewritten as

f (x) (x 1)(x 1) f2(x) (x 1)(x 1)(2x2 4x 2)

To find the remaining two zeros we can always use the Quadratic Formula.

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