Solve 8x^4-14x^2-9

$$8 x ^ { 4 } - 14 x ^ { 2 } - 9$$

Factor

$$\left(2x-3\right)\left(2x+3\right)\left(2x^{2}+1\right)$$

Solution Steps

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

$$8x^{4}-14x^{2}-9=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -9 and q divides the leading coefficient 8. List all candidates \frac{p}{q}.

$$±\frac{9}{8},±\frac{9}{4},±\frac{9}{2},±9,±\frac{3}{8},±\frac{3}{4},±\frac{3}{2},±3,±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$x=\frac{3}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 8x^{4}-14x^{2}-9 by 2\left(x-\frac{3}{2}\right)=2x-3 to get 4x^{3}+6x^{2}+2x+3. To factor the result, solve the equation where it equals to 0.

$$4x^{3}+6x^{2}+2x+3=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 3 and q divides the leading coefficient 4. List all candidates \frac{p}{q}.

$$±\frac{3}{4},±\frac{3}{2},±3,±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$x=-\frac{3}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 4x^{3}+6x^{2}+2x+3 by 2\left(x+\frac{3}{2}\right)=2x+3 to get 2x^{2}+1. To factor the result, solve the equation where it equals to 0.

$$2x^{2}+1=0$$

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

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

Do the calculations.

$$x=\frac{0±\sqrt{-8}}{4}$$

Polynomial 2x^{2}+1 is not factored since it does not have any rational roots.

$$2x^{2}+1$$

Rewrite the factored expression using the obtained roots.

$$\left(2x-3\right)\left(2x+3\right)\left(2x^{2}+1\right)$$

Evaluate

$$8x^{4}-14x^{2}-9$$