[Math] Factorization of a polynomial using synthetic division

Question

Factorize:
$$x^4-x^3-19x^2+49x-30$$

In the figure above, I have showed upto where I tried? Can anyone help me to complete it?

Answer 1

Now, suppose $G(x)=x^3-19x+30$ and note that $G(2)=0$.

So $(x-2)$ is a factor of $G(x)$ by the factor theorem.

By synthetic division, we get $$\color{red}{G(x)=(x-2)(x^2+2x-15)}$$

Again, suppose $H(x)=x^2+2x-15$ and note that $G(3)=0$.

So $(x-3)$ is a factor of $G(x)$ by the factor theorem.

By synthetic division, we get $$\color{orange}{H(x)=(x-3)(x+5)}$$

So we can write that $$\color{blue}{x^4-x^3-19x^2+49x-30=(x-1)(x-2)(x-3)(x+5)}$$