Solve Polynomialrootcalculator (4x^5-3x^3+2x^2-7) - Tiger Algebra

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Step 1 :

Equation at the end of step 1 :

((((4•(x5))-(3•(x3)))+(2•(x2)))-7)-((((x4)+2x3)-4x)-3)

Step 2 :

Equation at the end of step 2 :

((((4•(x5))-(3•(x3)))+2x2)-7)-(x4+2x3-4x-3)

Step 3 :

Equation at the end of step 3 :

((((4•(x5))-3x3)+2x2)-7)-(x4+2x3-4x-3)

Step 4 :

Equation at the end of step 4 :

(((22x5 - 3x3) + 2x2) - 7) - (x4 + 2x3 - 4x - 3)

Step 5 :

Trying to factor by pulling out :

5.1 Factoring: 4x5-x4-5x3+2x2+4x-4 Thoughtfully split the expression at hand into groups, each group having two terms :Group 1: 2x2-5x3 Group 2: 4x5-x4 Group 3: 4x-4 Pull out from each group separately :Group 1: (5x-2) • (-x2)Group 2: (4x-1) • (x4)Group 3: (x-1) • (4) Looking for common sub-expressions : Group 1: (5x-2) • (-x2) Group 3: (x-1) • (4) Group 2: (4x-1) • (x4)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 :

5.2 Find roots (zeroes) of : F(x) = 4x5-x4-5x3+2x2+4x-4Polynomial 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 integersThe 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 CoefficientIn this case, the Leading Coefficient is 4 and the Trailing Constant is -4. The factor(s) are: of the Leading Coefficient : 1,2 ,4 of the Trailing Constant : 1 ,2 ,4 Let us test ....

PQP/QF(P/Q)Divisor
-1 1 -1.00 -6.00
-1 2 -0.50 -5.06
-1 4 -0.25 -4.80
-2 1 -2.00 -108.00
-4 1 -4.00 -4020.00
1 1 1.00 0.00 x-1
1 2 0.50 -2.06
1 4 0.25 -2.95
2 1 2.00 84.00
4 1 4.00 3564.00

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms In our case this means that 4x5-x4-5x3+2x2+4x-4 can be divided with x-1

Polynomial Long Division :

5.3 Polynomial Long Division Dividing : 4x5-x4-5x3+2x2+4x-4 ("Dividend") By : x-1 ("Divisor")

dividend 4x5 - x4 - 5x3 + 2x2 + 4x - 4
- divisor * 4x4 4x5 - 4x4
remainder 3x4 - 5x3 + 2x2 + 4x - 4
- divisor * 3x3 3x4 - 3x3
remainder- 2x3 + 2x2 + 4x - 4
- divisor * -2x2 - 2x3 + 2x2
remainder 4x - 4
- divisor * 0x1
remainder 4x - 4
- divisor * 4x0 4x - 4
remainder0

Quotient : 4x4+3x3-2x2+4 Remainder: 0

Polynomial Roots Calculator :

5.4 Find roots (zeroes) of : F(x) = 4x4+3x3-2x2+4 See theory in step 5.2 In this case, the Leading Coefficient is 4 and the Trailing Constant is 4. The factor(s) are: of the Leading Coefficient : 1,2 ,4 of the Trailing Constant : 1 ,2 ,4 Let us test ....

PQP/QF(P/Q)Divisor
-1 1 -1.00 3.00
-1 2 -0.50 3.38
-1 4 -0.25 3.84
-2 1 -2.00 36.00
-4 1 -4.00 804.00
1 1 1.00 9.00
1 2 0.50 4.12
1 4 0.25 3.94
2 1 2.00 84.00
4 1 4.00 1188.00

Polynomial Roots Calculator found no rational roots

Final result :

(4x4 + 3x3 - 2x2 + 4) • (x - 1)

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