Question

H(x) = x4+ 2x3- 10x2 -18x+9

Answer 1

Let's group the cubic function: $h(x)=x^4+2x^3-10x^2-18x+9=(x-3)(x+3)(x^2+2x-1)$.

The first two roots are $\pm3$ however to find the last two roots we need to solve the square equation:

$x^2+2x-1=0 \\D=b^2-4ac=2^2-4\cdot1\cdot(-1)=8$. Now we know the discriminator, using the discriminator we're able to find the roots of the equation: $x_{1,2}=\frac{-b\pm\sqrt{D}}{2a}=\frac{-2\pm\sqrt{8}}{2}=\frac{-2\pm2\sqrt{2}}{2}=\frac{2(\sqrt{2}\pm1)}{2}$. The roots of the square equation are $x_{1}=-1-\sqrt{2}$ and $x_{2}=\sqrt{2}-1$.

The roots of the cubic function are $\pm3$, $-1-\sqrt{2}$ and $\sqrt{2}-1$.

Answer 2

H (x) = (x+3)(x-3)(2x+x^{2} -1)