solve : x^4-6x^3+13x^2-24x^1+36=0 given that the equation has multiple roots.
Answers
Given : x^4-6x^3+13x^2-24x^1+36=0
To Find : Roots
Solution:
x⁴ - 6x³ + 13x² - 24x + 36 = 0
x = 3
=> 3⁴ - 6.3³ + 13.3² - 24.3 + 36
= 81 - 162 + 117 - 72 + 36
= 234 - 234
= 0
x - 3 is a root
x⁴ - 6x³ + 13x² - 24x + 36 = (x - 3 ) (x³ - 3x² + 4x -12 )
(x - 3 ) (x³ - 3x² + 4x -12 ) = 0
=> x³ - 3x² + 4x -12 = 0
3³ - 3,3² + 4.3 -12 = 0
x -3 is again a root
=>x³ - 3x² + 4x -12 = (x - 3)(x² + 4)
(x² + 4) = 0
=> x² = - 4
=> x = ± 2i
roots are 3 , 3 , ± 2i
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Answer:
Step-by-step explanation:
Equation at the end of step 1
((((x4)-(6•(x3)))+13x2)-24x)+36 = 0
STEP
2
:
Equation at the end of step
2
:
((((x4) - (2•3x3)) + 13x2) - 24x) + 36 = 0
STEP
3
:
Polynomial Roots Calculator :
3.1 Find roots (zeroes) of : F(x) = x4-6x3+13x2-24x+36
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 36.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,9 ,12 ,18 ,36
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 80.00
-2 1 -2.00 200.00
-3 1 -3.00 468.00
-4 1 -4.00 980.00
-6 1 -6.00 3240.00
-9 1 -9.00 12240.00
-12 1 -12.00 33300.00
-18 1 -18.00 144648.00
-36 1 -36.00 1977300.00
1 1 1.00 20.00
2 1 2.00 8.00
3 1 3.00 0.00 x-3
4 1 4.00 20.00
6 1 6.00 360.00
9 1 9.00 3060.00
12 1 12.00 11988.00
18 1 18.00 73800.00
36 1 36.00 1415700.00
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
x4-6x3+13x2-24x+36
can be divided with x-3
Polynomial Long Division :
3.2 Polynomial Long Division
Dividing : x4-6x3+13x2-24x+36
("Dividend")
By : x-3 ("Divisor")
dividend x4 - 6x3 + 13x2 - 24x + 36
- divisor * x3 x4 - 3x3
remainder - 3x3 + 13x2 - 24x + 36
- divisor * -3x2 - 3x3 + 9x2
remainder 4x2 - 24x + 36
- divisor * 4x1 4x2 - 12x
remainder - 12x + 36
- divisor * -12x0 - 12x + 36
remainder 0
Quotient : x3-3x2+4x-12 Remainder: 0
Polynomial Roots Calculator :
3.3 Find roots (zeroes) of : F(x) = x3-3x2+4x-12
See theory in step 3.1
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 -40.00
-3 1 -3.00 -78.00
-4 1 -4.00 -140.00
-6 1 -6.00 -360.00
-12 1 -12.00 -2220.00
1 1 1.00 -10.00
2 1 2.00 -8.00
3 1 3.00 0.00 x-3
4 1 4.00 20.00
6 1 6.00 120.00
12 1 12.00 1332.00