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Step by Step Solution

Step 1 :

Equation at the end of step 1 :

  (((0 -  (x⁴)) +  7x²) -  x) -  17  = 0

Step 2 :

Step 3 :

Pulling out like terms :

3.1     Pull out like factors :

   -x⁴ + 7x² - x - 17  =

  -1 • (x⁴ - 7x² + x + 17)

Checking for a perfect cube :

3.2 x⁴ - 7x² + x + 17 is not a perfect cube

Trying to factor by pulling out :

3.3 Factoring: x⁴ - 7x² + x + 17

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: x + 17
Group 2: x⁴ - 7x²

Pull out from each group separately :

Group 1: (x + 17) • (1)
Group 2: (x² - 7) • (x²)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

3.4    Find roots (zeroes) of :       F(x) = x⁴ - 7x² + x + 17
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 1 and the Trailing Constant is 17.

The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,17

Let us test ....

P	Q	P/Q	F(P/Q)
-1	1	-1.00	10.00
-17	1	-17.00	81498.00
1	1	1.00	12.00
17	1	17.00	81532.00

Polynomial Roots Calculator found no rational roots

Equation at the end of step 3 :

  -x⁴ + 7x² - x - 17  = 0

Step 4 :

Quartic Equations :

4.1 Solve -x⁴+7x²-x-17 = 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