Solve x^5+5x^4+7x^3-x^2-8x-4

$$x ^ { 5 } + 5 x ^ { 4 } + 7 x ^ { 3 } - x ^ { 2 } - 8 x - 4$$

Factor

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

Solution Steps

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

$$x^{5}+5x^{4}+7x^{3}-x^{2}-8x-4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{5}+5x^{4}+7x^{3}-x^{2}-8x-4 by x-1 to get x^{4}+6x^{3}+13x^{2}+12x+4. To factor the result, solve the equation where it equals to 0.

$$x^{4}+6x^{3}+13x^{2}+12x+4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}+6x^{3}+13x^{2}+12x+4 by x+1 to get x^{3}+5x^{2}+8x+4. To factor the result, solve the equation where it equals to 0.

$$x^{3}+5x^{2}+8x+4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+5x^{2}+8x+4 by x+1 to get x^{2}+4x+4. To factor the result, solve the equation where it equals to 0.

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

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

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

Do the calculations.

$$x=\frac{-4±0}{2}$$

Solutions are the same.

$$x=-2$$

Rewrite the factored expression using the obtained roots.

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

Evaluate

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