Solve 8x^4+x^2-9

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

Factor

$$\left(x-1\right)\left(x+1\right)\left(8x^{2}+9\right)$$

Solution Steps

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

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

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 8x^{4}+x^{2}-9 by x-1 to get 8x^{3}+8x^{2}+9x+9. To factor the result, solve the equation where it equals to 0.

$$8x^{3}+8x^{2}+9x+9=0$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 8x^{3}+8x^{2}+9x+9 by x+1 to get 8x^{2}+9. To factor the result, solve the equation where it equals to 0.

$$8x^{2}+9=0$$

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 8 for a, 0 for b, and 9 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 8\times 9}}{2\times 8}$$

Do the calculations.

$$x=\frac{0±\sqrt{-288}}{16}$$

Polynomial 8x^{2}+9 is not factored since it does not have any rational roots.

$$8x^{2}+9$$

Rewrite the factored expression using the obtained roots.

$$\left(x-1\right)\left(x+1\right)\left(8x^{2}+9\right)$$

Evaluate

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