How to solve polynomial of degree 4?
I'm beginner in learning algebra and there is a question which came into my mind that how to find value of x in this type of equation -- >
$x^4 + x^3 + x = 3$
I know that one of its answer will be 1,but I was wondering how to solve it through equation form.
Please help.Thank you in advance.
2 Answers
$\begingroup$We have $$x^4+x^3+x=3\to x^4+x^3+x-3=0$$ $$\text{When we have powers of $x>3$, we usually find an easy solution by testing a few values.}$$ $$\text{For example, if we plug in $x=1$, we have $1+1+1-3=0$}$$ $$\text{This means that $x=1$ is one of the solutions, and that $(x-1)$ is factor.}$$
Now we could use synthetic division to further reduce our polynomial.
That will give us $x^3+2x^2+2x+3=0$
This is irrational roots, and so we cannot do much after this. Either you must use a calculator, use methods like Newton's method, or analyse the derivative and do it. There is no easy way after this.
$x=-1.81,1$
$\endgroup$ $\begingroup$So,
I'm assuming here that $x$ is integer because integer solution would be easier to find and work out.
$$x^4 + x^3 + x = 3$$
$$\implies x^3(x+1)+x=3$$
$$\implies x^3(x+1)+(x+1)=4$$
$$\implies (x^3+1)(x+1)=4$$
Here,
$$(x^3+1)(x+1)=2\times2$$
Thus, by comparing:
$x^3+1=2$, and, $x+1=2$.
So, $x=1$ is the only integral answer.
Also note that there might be other non-integral real answers.
