Enter the factors of the constant term and the leading coefficient into the calculator to determine the rational zeros of the polynomial equation; this calculator can also evaluate any of the variables given the others are known.

Rational Zero Theorem Formula

The following formula is used to calculate the rational zeros of a polynomial equation:

Z = pm frac{p}{q}

Variables:

  • Z is the rational zero
  • p is a factor of the constant term of the polynomial
  • q is a factor of the leading coefficient of the polynomial

To calculate the rational zeros of a polynomial equation, you need to find the factors of the constant term and the leading coefficient. The rational zeros are then obtained by taking the ratio of a factor of the constant term to a factor of the leading coefficient, considering both positive and negative values.

What is a Rational Zero Theorem?

The Rational Zero Theorem is a mathematical principle used to identify potential rational zeros (or roots) of a polynomial equation. According to this theorem, if a polynomial has integer coefficients, then every rational zero will have the form p/q, where p is a factor of the constant term (the last number) and q is a factor of the leading coefficient (the coefficient of the highest degree term).

This theorem provides a finite list of possible rational zeros, which can then be tested to determine the actual zeros of the polynomial. It is important to note that the Rational Zero Theorem only provides possible rational zeros, and does not guarantee that these values will indeed be zeros of the polynomial.

How to Calculate Rational Zero Theorem?

The following steps outline how to use the Rational Zero Theorem to find the rational zeros of a polynomial equation.

  1. First, write down the given polynomial equation in standard form.
  2. Next, list all the possible rational zeros of the equation. These are the numbers that can be obtained by dividing a factor of the constant term by a factor of the leading coefficient.
  3. Use synthetic division or long division to test each possible rational zero. Divide the polynomial equation by the possible zero and check if the remainder is zero.
  4. If the remainder is zero, then the tested number is a rational zero of the equation.
  5. Repeat steps 3 and 4 until all possible rational zeros have been tested.
  6. The rational zeros of the polynomial equation are the numbers that produced a remainder of zero when tested.

Example Problem:

Use the following polynomial equation to find the rational zeros:

f(x) = 2x^3 – 5x^2 + 3x – 2

Possible rational zeros:

±1, ±2

Testing possible zero 1:

Using synthetic division:

1 | 2 -5 3 -2
   |   2 -3  6
   |_________
     2 -3  6  4

The remainder is not zero, so 1 is not a rational zero.

Testing possible zero -1:

Using synthetic division:

-1 | 2 -5 3 -2
    |  -2  7 -10
    |_________
      2 -7 10 -12

The remainder is not zero, so -1 is not a rational zero.

Testing possible zero 2:

Using synthetic division:

2 | 2 -5 3 -2
   |   4 -2  2
   |_________
     2 -1  5  0

The remainder is zero, so 2 is a rational zero.

Therefore, the rational zeros of the polynomial equation are 2.