Question

Solve the equation x^4 -2x^3 +x = 380 please its a level 3 question please i need explanation dint give foolish answers​

Answer 1

$x^4-2x^3+x-380=0$

You can simply use the Rational Root theorem.

Possible roots = ± $\frac{(Factors-of- 380)}{(Factors-of-1)}$

Possible roots = ±(1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380)

f(1) ≠ 0

f(-1) ≠ 0

f(2) ≠ 0

f(-2) ≠ 0

f(-4) = 0

f(4) ≠ 0

f(5) = 0

Now, we know 2 of the roots.

Thus, $x^2 - x -20$ is a factor of $x^4-2x^3+x-380$

By dividing $x^4-2x^3+x-380$ by $x^2 - x -20$, we get:

$x^2-x+19$

By using the Quadratic Equation, we get the roots as:

$\frac{1+\sqrt{-75} }{2}$, $\frac{1-\sqrt{-75} }{2}$

or,

$\frac{1}{2}+\frac{5\sqrt{3}}{2}i$, $\frac{1}{2}-\frac{5\sqrt{3}}{2}i$

So the roots are: (-4), 5, $\frac{1}{2}+\frac{5\sqrt{3}}{2}i$ and  $\frac{1}{2}-\frac{5\sqrt{3}}{2}i$