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# Rational Zeros Calculator

Enter the function and click calculate button to calculate the actual rational roots using the rational zeros calculator

## Rational Zeros Calculator

Rational zeros calculator is used to find the actual rational roots of the given function. This possible rational zeros calculator evaluates the result with steps in a fraction of a second.

## What is the Rational zeros theorem?

Rational root theorem is a fundamental theorem in algebraic number theory and is used to determine the possible rational roots of a polynomial equation. It states that if a polynomial equation has a rational root, then that root must be expressible as a fraction p/q, where p is a divisor of the leading coefficient and q is a divisor of the constant term.

The theorem is important because it provides a way to simplify the process of finding the roots of a polynomial equation. For example, suppose we have a polynomial equation

**f(x) = ax ^{n }+ bx^{(n-1)} + ... + c**

The theorem states that any rational root of this equation must be of the form p/q, where p divides c and q divides a. This means that we can start by testing all the possible rational numbers of this form, instead of having to test every possible real number.

## How to find all the zeros of polynomials?

We have to follow some steps to find the zeros of a polynomial:

- List the factors of the constant term and the coefficient of the leading term.
- Now divide factors of the leadings with factors of the constant.
- Remove the duplicated terms.
- If we put the zeros in the polynomial, we get the remainder equal to zero.

## How to calculate rational zeros?

**Example:**

Evaluate the polynomial P(x)= 2x^{2 }- 5x - 3.

**Solution:**

**Step 1:** First we have to make the factors of constant 3 and leading coefficients 2.

Factors of 3 = +1, -1, 3, -3

Factors of 2 = +1, -1, 2, -2

**Step 2: **Divide the factors of the constant with the factors of the leading term and remove the duplicate terms.

Possible rational roots: 1/2, 1, 3/2, 3, -1, -3/2, -1/2, -3

**Check 1:** divide 2x^{2 }- 5x - 3 by x- 1.

P(1) = 2(1)^{2} - 5(1) - 3

P(1) = -6

**Check 2:** divide 2x^{2 }- 5x - 3 by x - 3.

P (3) = 2(3)^{2} - 5(3) - 3

P (3) = 0

**Check 3:** divide 2x^{2 }- 5x - 3 by x - 3/2.

P (3/2) = 2(3/2)^{2} - 5(3/2) - 3

P (3/2) = -6

**Check 4:** divide 2x^2-5x-3 by x+1/2.

P (-1/2) =2(-1/2)^2 -5(-1/2)-3

P (-1/2) = 0

**Check 5: **divide 2x^{2 }- 5x - 3 by x+3/2.

P (3/2) = 2(3/2)^{2} - 5(3/2) - 3

P (3/2) = 9

**Check 6: **divide 2x^{2 }- 5x - 3 by x- 1/2.

P (1/2) = 2(1/2)^{2 }- 5(1/2) - 3

P (1/2) =-5

**Check 7: **divide 2x^{2 }- 5x - 3 by x+3.

P (3) = 2(3)^{2} - 5(3) - 3

P (3) = 30

**Check 8:** divide 2x^{2 }- 5x - 3 by x+1.

P (-1) = 2(-1)^{2} - 5(-1) - 3

P (-1) = 5

**Actual rational roots are 3,-1/2**