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

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

Factor

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

Solution Steps

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

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

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

$$±3,±1$$

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

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

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

$$±3,±1$$

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

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

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, 4 for b, and 3 for c in the quadratic formula.

$$x=\frac{-4±\sqrt{4^{2}-4\times 1\times 3}}{2}$$

Do the calculations.

$$x=\frac{-4±2}{2}$$

Solve the equation x^{2}+4x+3=0 when ± is plus and when ± is minus.

$$x=-3 x=-1$$

Rewrite the factored expression using the obtained roots.

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

Evaluate

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