Question

Find all the possible rational roots of:
1. f(x)=2x^4-5x^3+8x^2+4x+7
2. f(x)=x^3+4x^2-3x-5
3. f(x)=x^3+25

Answer 1

1. Step 1 :

Simplify 2x4-5x3+8x2+4x + 7
Polynomial Roots Calculator :

1.1 Find roots (zeroes) of
F(x) = 2x4-5x3+8x2+4x+7
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 2 and the Trailing Constant is 7.

The factor(s) are:

of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,7

Let us test ....

P Q P/Q F(P/Q) Divisor
-1 1 -1.00 18.00
-1 2 -0.50 7.75
-7 1 -7.00 6888.00
-7 2 -3.50 605.50
1 1 1.00 16.00
1 2 0.50 10.50
7 1 7.00 3514.00
7 2 3.50 204.75

Polynomial Roots Calculator found no rational roots

Equation at the end of step 1 :

2x4 - 5x3 + 8x2 + 4x + 7 = 0
Step 2 :

Solve 2x4-5x3+8x2+4x+7 = 0
Quartic Equations :

2.1 Solve 2x4-5x3+8x2+4x+7 = 0

In search of an interavl at which the above polynomial changes sign, from negative to positive or the other wayaround.

Method of search: Calculate polynomial values for all integer points between x=-20 and x=+20

No interval at which a change of sign occures has been found. Consequently, Bisection Approximation can not be used. As this is a polynomial of an even degree it may not even have any real (as opposed to imaginary) roots

No Solutions found