Solve 3x^4+x^3+2x^2+4x-40

$$3 { x }^{ 4 } + { x }^{ 3 } +2 { x }^{ 2 } +4x-40$$

Factor

$$\left(3x-5\right)\left(x+2\right)\left(x^{2}+4\right)$$

Solution Steps

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

$$3x^{4}+x^{3}+2x^{2}+4x-40=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -40 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{40}{3},±40,±\frac{20}{3},±20,±\frac{10}{3},±10,±\frac{8}{3},±8,±\frac{5}{3},±5,±\frac{4}{3},±4,±\frac{2}{3},±2,±\frac{1}{3},±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{4}+x^{3}+2x^{2}+4x-40 by x+2 to get 3x^{3}-5x^{2}+12x-20. To factor the result, solve the equation where it equals to 0.

$$3x^{3}-5x^{2}+12x-20=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -20 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{20}{3},±20,±\frac{10}{3},±10,±\frac{5}{3},±5,±\frac{4}{3},±4,±\frac{2}{3},±2,±\frac{1}{3},±1$$

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.

$$x=\frac{5}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{3}-5x^{2}+12x-20 by 3\left(x-\frac{5}{3}\right)=3x-5 to get x^{2}+4. To factor the result, solve the equation where it equals to 0.

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

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 4 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{0±\sqrt{-16}}{2}$$

Polynomial x^{2}+4 is not factored since it does not have any rational roots.

$$x^{2}+4$$

Rewrite the factored expression using the obtained roots.

$$\left(3x-5\right)\left(x+2\right)\left(x^{2}+4\right)$$

Evaluate

$$\left(3x-5\right)\left(x+2\right)\left(x^{2}+4\right)$$