Question

Find the square root of x^4 - 6x^3 + 13x^2 - 12x + 4

Answer 1

x4-6x3+13x2-14x+6=0

Four solutions were found :

x = 3
x = 1
x =(2-√-4)/2=1-i= 1.0000-1.0000i
x =(2+√-4)/2=1+i= 1.0000+1.0000i
Step by step solution :

Step 1 :

Equation at the end of step 1 :

((((x4)-(6•(x3)))+13x2)-14x)+6 = 0
Step 2 :

Equation at the end of step 2 :

((((x4) - (2•3x3)) + 13x2) - 14x) + 6 = 0
Step 3 :

Polynomial Roots Calculator :

3.1 Find roots (zeroes) of : F(x) = x4-6x3+13x2-14x+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 1 and the Trailing Constant is 6.

The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,6

Let us test ....

P Q P/Q F(P/Q) Divisor
-1 1 -1.00 40.00
-2 1 -2.00 150.00
-3 1 -3.00 408.00
-6 1 -6.00 3150.00
1 1 1.00 0.00 x-1
2 1 2.00 -2.00
3 1 3.00 0.00 x-3
6 1 6.00 390.00

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+13x2-14x+6
can be divided by 2 different polynomials,including by x-3

Polynomial Long Division :

3.2 Polynomial Long Division
Dividing : x4-6x3+13x2-14x+6
("Dividend")
By : x-3 ("Divisor")

dividend x4 - 6x3 + 13x2 - 14x + 6
- divisor * x3 x4 - 3x3
remainder - 3x3 + 13x2 - 14x + 6
- divisor * -3x2 - 3x3 + 9x2
remainder 4x2 - 14x + 6
- divisor * 4x1 4x2 - 12x
remainder - 2x + 6
- divisor * -2x0 - 2x + 6
remainder 0
Quotient : x3-3x2+4x-2 Remainder: 0

Polynomial Roots Calculator :

3.3 Find roots (zeroes) of : F(x) = x3-3x2+4x-2

See theory in step 3.1
In this case, the Leading Coefficient is 1 and the Trailing Constant is -2.

The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2

Let us test ....

P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -10.00
-2 1 -2.00 -30.00
1 1 1.00 0.00 x-1
2 1 2.00 2.00

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+4x-2
can be divided with x-1

Polynomial Long Division :

3.4 Polynomial Long Division
Dividing : x3-3x2+4x-2
("Dividend")
By : x-1 ("Divisor")

dividend x3 - 3x2 + 4x - 2
- divisor * x2 x3 - x2
remainder - 2x2 + 4x - 2
- divisor * -2x1 - 2x2 + 2x
remainder 2x - 2
- divisor * 2x0 2x - 2
remainder 0
Quotient : x2-2x+2 Remainder: 0

Trying to factor by splitting the middle term

3.5 Factoring x2-2x+2

The first term is, x2 its coefficient is 1 .
The middle term is, -2x its coefficient is -2 .
The last term, "the constant", is +2

Step-1 : Multiply the coefficient of the first term by the constant 1 • 2 = 2

Step-2 : Find two factors of 2 whose sum

Answer 2

Answer:

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