Solve 2x^4+3x^3-8x^2-9x+6

$$2 x ^ { 4 } + 3 x ^ { 3 } - 8 x ^ { 2 } - 9 x + 6$$

Factor

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

Solution Steps

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

$$2x^{4}+3x^{3}-8x^{2}-9x+6=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 6 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±3,±6,±\frac{3}{2},±1,±2,±\frac{1}{2}$$

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.

$$x=-2$$

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

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

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

$$±\frac{3}{2},±3,±\frac{1}{2},±1$$

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.

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

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

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

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 -3 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\left(-3\right)}}{2}$$

Do the calculations.

$$x=\frac{0±2\sqrt{3}}{2}$$

Solve the equation x^{2}-3=0 when ± is plus and when ± is minus.

$$x=-\sqrt{3} x=\sqrt{3}$$

Rewrite the factored expression using the obtained roots. Polynomial x^{2}-3 is not factored since it does not have any rational roots.

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

Evaluate

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