Finding Zeros Essay

Finding zeros

The Fundamental Theorem of Algebra

A polynomial function f(x) of degree n (where n>0) has n complex solutions fro the equation f(x)=0. Thus if a polynomials largest exponent is 4, the degree of that polynomial is 4. The degree does not tell us what the solutions are, but the number of solutions a polynomial has.

The Rational Zero Theorem

If has integer coefficients and (where is reduced to lowest terms) is a rational zero of f, then p is a factor of the constant term, a0, and q is a factor of the leading coefficient, an. We can use the rational zero theorem to find all the rational zeros of a polynomial. For example:

f(x)= x4 – 4x3 – 5x2 + 16x + 4

* First arrange the function in descending order.

Identify the constant term, in this case 4. These are all the possible values of p.

And the leading coefficient, in this case 1. These are all the possible values of q.

* Find the factors of both:

* Factors of 4 : 1, -1, 2, -2, 4, -4

* Factors of 1: 1, -1,

* Next, set the factors of p, 4, over the factors of q, 1:

* Factor of13Factors of 1=±1,±2, ±4±1

* Simplify and eliminate any duplicate.

* Thus, the possible rational zeros of x4 – 4x3 – 5x2 + 16x + 4 are: ±1,±2, ±4

Synthetic Division

Next, synthetic division is used to test the all possible rational zeros and find the zeros of the function. We bring down the first number, multiply it by two and add or subtract as needed from the next number. We continue doing this with all of them.

2 1 -4 -5 16 4 2 -4 -18-4 1 -2 -9 -2 0

The zero remainder indicates that 2 is a root of x4 – 4x3 – 2x2 + 16x + 4=0. Since 2 is a root of the equation, x-2 is a factor of the polynomial. The first four numbers in the bottom row of the division give the coefficients of the second factor. This means that x4 – 4x3 – 5x2 + 16x + 4 = (x-2) (x 3-2x2-9x-2) = 0

We can look for rational roots of the polynomial...