Solve 5x^4-3x^2-2

$$5 { x }^{ 4 } -3 { x }^{ 2 } -2$$

Factor

$$\left(x-1\right)\left(x+1\right)\left(5x^{2}+2\right)$$

Solution Steps

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

$$5x^{4}-3x^{2}-2=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 5. List all candidates \frac{p}{q}.

$$±\frac{2}{5},±2,±\frac{1}{5},±1$$

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.

$$x=1$$

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

$$5x^{3}+5x^{2}+2x+2=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 2 and q divides the leading coefficient 5. List all candidates \frac{p}{q}.

$$±\frac{2}{5},±2,±\frac{1}{5},±1$$

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.

$$x=-1$$

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

$$5x^{2}+2=0$$

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 5 for a, 0 for b, and 2 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 5\times 2}}{2\times 5}$$

Do the calculations.

$$x=\frac{0±\sqrt{-40}}{10}$$

Polynomial 5x^{2}+2 is not factored since it does not have any rational roots.

$$5x^{2}+2$$

Rewrite the factored expression using the obtained roots.

$$\left(x-1\right)\left(x+1\right)\left(5x^{2}+2\right)$$

Evaluate

$$5x^{4}-3x^{2}-2$$