A calculator to calculate the real and complex zeros of a polynomial is presented.

\( a \) is a zero of a polynomial \( P(x) \) if and only if \( P(a) = 0 \)

or

\( a \) is a zero of a polynomial \( P(x) \) if and only if \( x - a \) is a factor of \( P(x) \)

Note that the zeros of the polynomial \( P(x) \) refer to the values of \( x \) that makes \( P(x) \) equal to zero. But both the zeros and the roots of a polynomial are found using factoring and the factor theorem [1 2].

Example

Find the zeros of the polynomial \( P(x) = x^2 + 5x - 14 \).

Solution

Factor \( P(x) \) as follows

\( P(x) = (x-2)(x+7) \)

Set \( P(x) = 0 \) and solve

\( P(x) = (x-2)(x+7) = 0 \)

Apply the factor theorem [1 2] and write that each factor is equal to zero.

\( x-2 = 0 \) or \( x+7 = 0 \)

Solve to obtain

\( x = 2 \) and \( x = - 7 \)

Hence the zeros of \( P(x) \) are \( x = 2 \) and \( x = - 7 \)

1 - Enter and edit polynomial \( P(x) \) and click "Enter Polynomial" then check what you have entered and edit if needed.

Note that the five operators used are: + (plus) , - (minus), , ^ (power) and * (multiplication). (example: P(x) = -2*x^4+8*x^3+14*x^2-44*x-48).(more notes on editing functions are located below)

2 - Click "Calculate Zeros" to obain the zeros of the polynomial.

Note that the zeros of some polynomials take a large amount of time to be computated and their expressions may be quite complicated to understand.

Notes: In editing functions, use the following:

1 - The five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: P(x) = 2*x^2 - 2*x - 4 )

Here are some examples of polynomials that you may copy and paste to practice:

x^2 - 9 x^2 + 9 x^2 + 2*x + 7 x^3 + 2*x - 3 3*x^4 - 3

x^5+5*x^4+3*x^3+x^2-10*x-120 x^5+4x^4-7x^3-28x^2+6x+24

x^4 - 4*x^3 + 3 (this one has very complicated zeros and takes time to compute; try it to have an idea.)

polynomials

Factor Polynomials

Find Zeros of Polynomials

Algebra and Trigonometry - Swokowsky Cole - 1997 - ISBN: 0-534-95308-5

Algebra and Trigonometry with Analytic Geometry - R.E.Larson , R.P. Hostetler , B.H. Edwards, D.E. Heyd - 1997 - ISBN: 0-669-41723-8