Solve {l}{4x^4+7x^2}{-2}

$$\left. \begin{array} { l } { 4 x ^ { 4 } + 7 x ^ { 2 } } \\ { - 2 } \end{array} \right.$$

Factor

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

Solution Steps

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

$$4x^{4}+7x^{2}-2=0$$

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 4. List all candidates \frac{p}{q}.

$$±\frac{1}{2},±1,±2,±\frac{1}{4}$$

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.

$$x=\frac{1}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 4x^{4}+7x^{2}-2 by 2\left(x-\frac{1}{2}\right)=2x-1 to get 2x^{3}+x^{2}+4x+2. To factor the result, solve the equation where it equals to 0.

$$2x^{3}+x^{2}+4x+2=0$$

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 2. List all candidates \frac{p}{q}.

$$±1,±2,±\frac{1}{2}$$

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.

$$x=-\frac{1}{2}$$

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

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

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

$$x=\frac{0±\sqrt{0^{2}-4\times 1\times 2}}{2}$$

Do the calculations.

$$x=\frac{0±\sqrt{-8}}{2}$$

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

$$x^{2}+2$$

Rewrite the factored expression using the obtained roots.

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

Evaluate

$$4x^{4}+7x^{2}-2$$