Step 1 :Equation at the end of step 1
2•(x4))+(3•(x3)))-2x2)-3x)-6)•(2x+3)
Step 2 :
Equation at the end of step 2 :
2•(x4))+3x3)-2x2)-3x)-6)•(2x+3)
Step 3 :
Equation at the end of step 3 :
2x4 + 3x3) - 2x2) - 3x) - 6) • (2x + 3)
Step 4 :
Polynomial Roots Calculator :
4.1 Find roots (zeroes) of : F(x) = 2x4+3x3-2x2-3x-6
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 -6.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -6.00
-1 2 -0.50 -5.25
-2 1 -2.00 0.00 x+2
-3 1 -3.00 66.00
-3 2 -1.50 -6.00
-6 1 -6.00 1884.00
1 1 1.00 -6.00
1 2 0.50 -7.50
2 1 2.00 36.00
3 1 3.00 210.00
3 2 1.50 5.25
6 1 6.00 3144.00
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
2x4+3x3-2x2-3x-6
can be divided with x+2
Polynomial Long Division :
4.2 Polynomial Long Division
Dividing : 2x4+3x3-2x2-3x-6
("Dividend")
By : x+2 ("Divisor")
dividend 2x4 + 3x3 - 2x2 - 3x - 6
- divisor * 2x3 2x4 + 4x3
remainder - x3 - 2x2 - 3x - 6
- divisor * -x2 - x3 - 2x2
remainder - 3x - 6
- divisor * 0x1
remainder - 3x - 6
- divisor * -3x0 - 3x - 6
remainder 0
Quotient : 2x3-x2-3 Remainder: 0
Polynomial Roots Calculator :
4.3 Find roots (zeroes) of : F(x) = 2x3-x2-3
See theory in step 4.1
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
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -6.00
-1 2 -0.50 -3.50
-3 1 -3.00 -66.00
-3 2 -1.50 -12.00
1 1 1.00 -2.00
1 2 0.50 -3.00
3 1 3.00 42.00
3 2 1.50 1.50
Polynomial Roots Calculator found no rational roots
Final result :
(2x3 - x2 - 3) • (x + 2) • (2x + 3)
