Solve 3x^4+6x^3-2x^2-10x-5

$$3 { x }^{ 4 } +6 { x }^{ 3 } -2 { x }^{ 2 } -10x-5$$

Factor

$$\left(3x^{2}-5\right)\left(x+1\right)^{2}$$

Solution Steps

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

$$3x^{4}+6x^{3}-2x^{2}-10x-5=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -5 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{5}{3},±5,±\frac{1}{3},±1$$

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 3x^{4}+6x^{3}-2x^{2}-10x-5 by x+1 to get 3x^{3}+3x^{2}-5x-5. To factor the result, solve the equation where it equals to 0.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -5 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{5}{3},±5,±\frac{1}{3},±1$$

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 3x^{3}+3x^{2}-5x-5 by x+1 to get 3x^{2}-5. To factor the result, solve the equation where it equals to 0.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\times 3\left(-5\right)}}{2\times 3}$$

Do the calculations.

$$x=\frac{0±2\sqrt{15}}{6}$$

Solve the equation 3x^{2}-5=0 when ± is plus and when ± is minus.

$$x=-\frac{\sqrt{15}}{3} x=\frac{\sqrt{15}}{3}$$

Rewrite the factored expression using the obtained roots. Polynomial 3x^{2}-5 is not factored since it does not have any rational roots.

$$\left(3x^{2}-5\right)\left(x+1\right)^{2}$$

Evaluate

$$\left(3x^{2}-5\right)\left(x+1\right)^{2}$$