Solve u^4-5u^2+4

$$u ^ { 4 } - 5 u ^ { 2 } + 4$$

Factor

$$\left(u-2\right)\left(u-1\right)\left(u+1\right)\left(u+2\right)$$

Solution Steps

To factor the expression, solve the equation where it equals to 0.

$$u^{4}-5u^{2}+4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$u=1$$

By Factor theorem, u-k is a factor of the polynomial for each root k. Divide u^{4}-5u^{2}+4 by u-1 to get u^{3}+u^{2}-4u-4. To factor the result, solve the equation where it equals to 0.

$$u^{3}+u^{2}-4u-4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$u=-1$$

By Factor theorem, u-k is a factor of the polynomial for each root k. Divide u^{3}+u^{2}-4u-4 by u+1 to get u^{2}-4. To factor the result, solve the equation where it equals to 0.

$$u^{2}-4=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 0 for b, and -4 for c in the quadratic formula.

$$u=\frac{0±\sqrt{0^{2}-4\times 1\left(-4\right)}}{2}$$

Do the calculations.

$$u=\frac{0±4}{2}$$

Solve the equation u^{2}-4=0 when ± is plus and when ± is minus.

$$u=-2 u=2$$

Rewrite the factored expression using the obtained roots.

$$\left(u-2\right)\left(u-1\right)\left(u+1\right)\left(u+2\right)$$

Evaluate

$$u^{4}-5u^{2}+4$$