Answer:
(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
