Solve Polynomialrootcalculator (-2x^6+5x^5-3x^4-6x^3+12x^2+5x-3)

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Step 1 :

Equation at the end of step 1 :

Step 2 :

Equation at the end of step 2 :

Step 3 :

Equation at the end of step 3 :

Step 4 :

Equation at the end of step 4 :

Step 5 :

Equation at the end of step 5 :

Step 6 :

-2x6 + 5x5 - 3x4 - 6x3 + 12x2 + 5x - 3 Simplify —————————————————————————————————————— -x3 + x2 - x - 3

Step 7 :

Pulling out like terms :

7.1 Pull out like factors : -2x6 + 5x5 - 3x4 - 6x3 + 12x2 + 5x - 3 = -1 • (2x6 - 5x5 + 3x4 + 6x3 - 12x2 - 5x + 3)

Step 8 :

Pulling out like terms :

8.1 Pull out like factors : -x3 + x2 - x - 3 = -1 • (x3 - x2 + x + 3)

Checking for a perfect cube :

8.2 x3 - x2 + x + 3 is not a perfect cube

Trying to factor by pulling out :

8.3 Factoring: x3 - x2 + x + 3 Thoughtfully split the expression at hand into groups, each group having two terms :Group 1: x + 3 Group 2: x3 - x2 Pull out from each group separately :Group 1: (x + 3) • (1)Group 2: (x - 1) • (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 :

8.4 Find roots (zeroes) of : F(x) = 2x6 - 5x5 + 3x4 + 6x3 - 12x2 - 5x + 3Polynomial 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 2 and the Trailing Constant is 3. The factor(s) are: of the Leading Coefficient : 1,2 of the Trailing Constant : 1 ,3 Let us test ....

PQP/QF(P/Q)Divisor
-1 1 -1.00 0.00 x + 1
-1 2 -0.50 2.12
-3 1 -3.00 2664.00
-3 2 -1.50 39.19
1 1 1.00 -8.00
1 2 0.50 -1.69
3 1 3.00 528.00
3 2 1.50 -11.25

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 2x6 - 5x5 + 3x4 + 6x3 - 12x2 - 5x + 3 can be divided with x + 1

Polynomial Long Division :

8.5 Polynomial Long Division Dividing : 2x6 - 5x5 + 3x4 + 6x3 - 12x2 - 5x + 3 ("Dividend") By : x + 1 ("Divisor")

dividend 2x6 - 5x5 + 3x4 + 6x3 - 12x2 - 5x + 3
- divisor * 2x5 2x6 + 2x5
remainder- 7x5 + 3x4 + 6x3 - 12x2 - 5x + 3
- divisor * -7x4 - 7x5 - 7x4
remainder 10x4 + 6x3 - 12x2 - 5x + 3
- divisor * 10x3 10x4 + 10x3
remainder- 4x3 - 12x2 - 5x + 3
- divisor * -4x2 - 4x3 - 4x2
remainder- 8x2 - 5x + 3
- divisor * -8x1 - 8x2 - 8x
remainder 3x + 3
- divisor * 3x0 3x + 3
remainder0

Quotient : 2x5-7x4+10x3-4x2-8x+3 Remainder: 0

Polynomial Roots Calculator :

8.6 Find roots (zeroes) of : F(x) = 2x5-7x4+10x3-4x2-8x+3 See theory in step 8.4 In this case, the Leading Coefficient is 2 and the Trailing Constant is 3. The factor(s) are: of the Leading Coefficient : 1,2 of the Trailing Constant : 1 ,3 Let us test ....

PQP/QF(P/Q)Divisor
-1 1 -1.00 -12.00
-1 2 -0.50 4.25
-3 1 -3.00 -1332.00
-3 2 -1.50 -78.38
1 1 1.00 -4.00
1 2 0.50 -1.12
3 1 3.00 132.00
3 2 1.50 -4.50

Polynomial Roots Calculator found no rational roots

Polynomial Roots Calculator :

8.7 Find roots (zeroes) of : F(x) = x3-x2+x+3 See theory in step 8.4 In this case, the Leading Coefficient is 1 and the Trailing Constant is 3. The factor(s) are: of the Leading Coefficient : 1 of the Trailing Constant : 1 ,3 Let us test ....

PQP/QF(P/Q)Divisor
-1 1 -1.00 0.00 x+1
-3 1 -3.00 -36.00
1 1 1.00 4.00
3 1 3.00 24.00

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 x3-x2+x+3 can be divided with x+1

Polynomial Long Division :

8.8 Polynomial Long Division Dividing : x3-x2+x+3 ("Dividend") By : x+1 ("Divisor")

dividend x3 - x2 + x + 3
- divisor * x2 x3 + x2
remainder- 2x2 + x + 3
- divisor * -2x1 - 2x2 - 2x
remainder 3x + 3
- divisor * 3x0 3x + 3
remainder0

Quotient : x2-2x+3 Remainder: 0

Trying to factor by splitting the middle term

8.9 Factoring x2-2x+3 The first term is, x2 its coefficient is 1 .The middle term is, -2x its coefficient is -2 .The last term, "the constant", is +3 Step-1 : Multiply the coefficient of the first term by the constant 1 • 3 = 3 Step-2 : Find two factors of 3 whose sum equals the coefficient of the middle term, which is -2 .

-3 + -1 = -4
-1 + -3 = -4
1 + 3 = 4
3 + 1 = 4

Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored

Canceling Out :

8.10 Cancel out (x+1) which appears on both sides of the fraction line.

Step 9 :

Pulling out like terms :

9.1 Pull out like factors : -x2 + 2x - 3 = -1 • (x2 - 2x + 3)

Polynomial Roots Calculator :

9.2 Find roots (zeroes) of : F(x) = -2x5 + 7x4 - 10x3 + 4x2 + 8x - 3 See theory in step 8.4 In this case, the Leading Coefficient is -2 and the Trailing Constant is -3. The factor(s) are: of the Leading Coefficient : 1,2 of the Trailing Constant : 1 ,3 Let us test ....

PQP/QF(P/Q)Divisor
-1 1 -1.00 12.00
-1 2 -0.50 -4.25
-3 1 -3.00 1332.00
-3 2 -1.50 78.38
1 1 1.00 4.00
1 2 0.50 1.12
3 1 3.00 -132.00
3 2 1.50 4.50

Polynomial Roots Calculator found no rational roots

Final result :

-2x5 + 7x4 - 10x3 + 4x2 + 8x - 3 ———————————————————————————————— -x2 + 2x - 3

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