Finding The Roots Of Polynomials - Tiger Algebra

Step by step solution :

Step 1 :

Equation at the end of step 1 :

(((3•(x7))-(2•(x5)))+3x3)-4 = 0

Step 2 :

Equation at the end of step 2 :

(((3 • (x7)) - 2x5) + 3x3) - 4 = 0

Step 3 :

Equation at the end of step 3 :

((3x7 - 2x5) + 3x3) - 4 = 0

Step 4 :

Checking for a perfect cube :

4.1 3x7-2x5+3x3-4 is not a perfect cube

Trying to factor by pulling out :

4.2 Factoring: 3x7-2x5+3x3-4 Thoughtfully split the expression at hand into groups, each group having two terms :Group 1: 3x3-4 Group 2: 3x7-2x5 Pull out from each group separately :Group 1: (3x3-4) • (1)Group 2: (3x2-2) • (x5)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 :

4.3 Find roots (zeroes) of : F(x) = 3x7-2x5+3x3-4Polynomial 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 3 and the Trailing Constant is -4. The factor(s) are: of the Leading Coefficient : 1,3 of the Trailing Constant : 1 ,2 ,4 Let us test ....

P	Q	P/Q	F(P/Q)	Divisor
-1	1	-1.00	-8.00
-1	3	-0.33	-4.10
-2	1	-2.00	-348.00
-2	3	-0.67	-4.80
-4	1	-4.00	-47300.00
-4	3	-1.33	-25.16
1	1	1.00	0.00	x-1
1	3	0.33	-3.90
2	1	2.00	340.00
2	3	0.67	-3.20
4	1	4.00	47292.00
4	3	1.33	17.16

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 3x7-2x5+3x3-4 can be divided with x-1

Polynomial Long Division :

4.4 Polynomial Long Division Dividing : 3x7-2x5+3x3-4 ("Dividend") By : x-1 ("Divisor")

dividend	3x7	-	2x5	+	3x3	-	4
- divisor	* 3x6	3x7	-	3x6
remainder	3x6	-	2x5	+	3x3	-	4
- divisor	* 3x5	3x6	-	3x5
remainder	x5	+	3x3	-	4
- divisor	* x4	x5	-	x4
remainder	x4	+	3x3	-	4
- divisor	* x3	x4	-	x3
remainder	4x3	-	4
- divisor	* 4x2	4x3	-	4x2
remainder	4x2	-	4
- divisor	* 4x1	4x2	-	4x
remainder	4x	-	4
- divisor	* 4x0	4x	-	4
remainder	0

Quotient : 3x6+3x5+x4+x3+4x2+4x+4 Remainder: 0

Polynomial Roots Calculator :

4.5 Find roots (zeroes) of : F(x) = 3x6+3x5+x4+x3+4x2+4x+4 See theory in step 4.3 In this case, the Leading Coefficient is 3 and the Trailing Constant is 4. The factor(s) are: of the Leading Coefficient : 1,3 of the Trailing Constant : 1 ,2 ,4 Let us test ....

P	Q	P/Q	F(P/Q)	Divisor
-1	1	-1.00	4.00
-1	3	-0.33	3.08
-2	1	-2.00	116.00
-2	3	-0.67	2.88
-4	1	-4.00	9460.00
-4	3	-1.33	10.78
1	1	1.00	20.00
1	3	0.33	5.84
2	1	2.00	340.00
2	3	0.67	9.60
4	1	4.00	15764.00
4	3	1.33	51.47

Polynomial Roots Calculator found no rational roots

Equation at the end of step 4 :

(3x6 + 3x5 + x4 + x3 + 4x2 + 4x + 4) • (x - 1) = 0

Step 5 :

Theory - Roots of a product :

5.1 A product of several terms equals zero.When a product of two or more terms equals zero, then at least one of the terms must be zero.We shall now solve each term = 0 separatelyIn other words, we are going to solve as many equations as there are terms in the productAny solution of term = 0 solves product = 0 as well.

Equations of order 5 or higher :

5.2 Solve 3x6+3x5+x4+x3+4x2+4x+4 = 0In 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

Solving a Single Variable Equation :

5.3 Solve : x-1 = 0Add 1 to both sides of the equation : x = 1

One solution was found :

x = 1