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(x+3)(x+2)=0

x = -3

x = -2

x =  0.0000 + 1.0000 i

Step-by-step explanation:

x^4 - 5x^3 +7x^2-5x+6 =0

We need to find the factors for 6 when multiplied together they give us 6

And when they are added they add up to 5

(x+3)(x+2)=0

x = -3

x = -2

x =  0.0000 + 1.0000 i

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

 of -x5/ x3-2x2+x-2 and of -x5 / x^2- 2x2 and  x^4-2x2 2x2

Which also separates 7 the same x7- 2x2+x-2 +x-2  and again to make (6 of the 7 )is 2x2 + x  then +x to make 7

The first one can be divided with  x-2 +1

Proof

3.5    Find roots (zeroes) of :       F(x) = x2+1

3.4    Polynomial Long Division

Dividing :  x3-2x2+x-2

                             ("Dividend")

By         :    x-2    ("Divisor")

dividend     x3  -  2x2  +  x  -  2

- divisor  * x2     x3  -  2x2        

remainder             x  -  2

- divisor  * 0x1                

remainder             x  -  2

- divisor  * x0             x  -  2

remainder                0

Quotient :  x2+1  Remainder:  0

    See theory in step 3.1

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        24.00    

     -2       1        -2.00        100.00    

     -3       1        -3.00        300.00    

     -6       1        -6.00        2664.00    

     1       1        1.00        4.00    

     2       1        2.00        0.00      x-2

     3       1        3.00        0.00      x-3

     6       1        6.00        444.00  

the function makes a strait line horizontally with no x intercept but a y intercept at 9.