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3x^{4}+x^{3}+2x^{2}+4x-40=0
To factor the expression, solve the equation where it equals to 0.
±\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
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}.
x=-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.
3x^{3}-5x^{2}+12x-20=0
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.
±\frac{20}{3},±20,±\frac{10}{3},±10,±\frac{5}{3},±5,±\frac{4}{3},±4,±\frac{2}{3},±2,±\frac{1}{3},±1
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}.
x=\frac{5}{3}
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}+4=0
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=\frac{0±\sqrt{0^{2}-4\times 1\times 4}}{2}
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{-16}}{2}
Do the calculations.
x^{2}+4
Polynomial x^{2}+4 is not factored since it does not have any rational roots.
\left(3x-5\right)\left(x+2\right)\left(x^{2}+4\right)
Rewrite the factored expression using the obtained roots.