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?

The 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) = axⁿ + 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)= 2x² - 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 2x² - 5x - 3 by x- 1.

P(1) = 2(1)² - 5(1) - 3

P(1) = -6

Check 2: Divide 2x² - 5x - 3 by x - 3.

P (3) = 2(3)² - 5(3) - 3

P (3) = 0 

Check 3: Divide 2x² - 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 2x² - 5x - 3 by x+3/2.

P (3/2) = 2(3/2)² - 5(3/2) - 3

P (3/2) = 9

Check 6: Divide 2x² - 5x - 3 by x- 1/2.

P (1/2) = 2(1/2)² - 5(1/2) - 3

P (1/2) =-5

Check 7: Divide 2x² - 5x - 3 by x+3.

P (3) = 2(3)² - 5(3) - 3

P (3) = 30

Check 8: Divide 2x² - 5x - 3 by x+1.

P (-1) = 2(-1)² - 5(-1) - 3

P (-1) = 5

Actual rational roots are 3,-1/2