Question

 Find all rational solutions of x⁴−6x³+22x²−30x+13=0

Answer 1

$x^4−6x^3+22x^2−30x+13=0$

Let f(x) = x⁴−6x³+22x²−30x+13

$f(1) = 1-6+22-30+13=0$

So, $(x-1)$ is a factor of $f(x).$

So, $x^4−6x^3+22x^2−30x+13 = (x-1)$ ($x^3-5x^2+17x-13$ )

Let $g(x)=x^3-5x^2+17x-13$

$g(1)=1-5+17-13=0$

So, $(x-1)$ is a factor of $g(x).$

Thus, $f(x)=(x-1).g(x)=(x-1)(x-1)(x^2-4x+13)$

Now, put $(x^2-4x+13)=0$

Using Quadratic formula:

$x=\frac{4\pm\sqrt{16-52}}{2}=\frac{4\pm6i}{2}=2\pm3i$

Hence, the solution are $x=1,1,2+3i,2-3i$ and rational solutions are $x=1,1$