x^4-1; x^3-11x^2+x-11
Answers
Step-by-step explanation:
Final result :
(x2 + 1) • (x + 3) • (x - 4)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((((x4)-(x3))-11x2)-x)-12
:Final result :
(x2 + 1) • (x + 3) • (x - 4)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((((x4)-(x3))-11x2)-x)-12
Step 2 :
Polynomial Roots Calculator :
2.1 Find roots (zeroes) of : F(x) = x4-x3-11x2-x-12
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 -12.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,12
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -20.00
-2 1 -2.00 -30.00
-3 1 -3.00 0.00 x+3
-4 1 -4.00 136.00
-6 1 -6.00 1110.00
-12 1 -12.00 20880.00
1 1 1.00 -24.00
2 1 2.00 -50.00
3 1 3.00 -60.00
4 1 4.00 0.00 x-4
6 1 6.00 666.00
12 1 12.00 17400.00
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
x4-x3-11x2-x-12
can be divided by 2 different polynomials,including by x-4
Polynomial Long Division :
2.2 Polynomial Long Division
Dividing : x4-x3-11x2-x-12
("Dividend")
By : x-4 ("Divisor")
dividend x4 - x3 - 11x2 - x - 12
- divisor * x3 x4 - 4x3
remainder 3x3 - 11x2 - x - 12
- divisor * 3x2 3x3 - 12x2
remainder x2 - x - 12
- divisor * x1 x2 - 4x
remainder 3x - 12
- divisor * 3x0 3x - 12
remainder 0
Quotient : x3+3x2+x+3 Remainder: 0
Polynomial Roots Calculator :
2.3 Find roots (zeroes) of : F(x) = x3+3x2+x+3
See theory in step 2.1
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 ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 4.00
-3 1 -3.00 0.00 x+3
1 1 1.00 8.00
3 1 3.00 60.00
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+3x2+x+3
can be divided with x+3
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3+3x2+x+3
("Dividend")
By : x+3 ("Divisor")
dividend x3 + 3x2 + x + 3
- divisor * x2 x3 + 3x2
remainder x + 3
- divisor * 0x1
remainder x + 3
- divisor * x0 x + 3
remainder 0
Quotient : x2+1 Remainder: 0
Polynomial Roots Calculator :
2.5 Find roots (zeroes) of : F(x) = x2+1
See theory in step 2.1
In this case, the Leading Coefficient is 1 and the Trailing Constant is 1.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 2.00
1 1 1.00 2.00
Polynomial Roots Calculator found no rational roots
Final result :
(x2 + 1) • (x + 3) • (x - 4)
Processing ends successfully
Step-by-step explanation: