Solve P(x)=x^{4}+4x^3-x^3-4x^2+x^2+4x-x-4

$$P ( x ) = x ^ { 4 } + 4 x ^ { 3 } - x ^ { 3 } - 4 x ^ { 2 } + x ^ { 2 } + 4 x - x - 4$$

Evaluate

$$\left(x-1\right)\left(x+4\right)\left(x^{2}+1\right)$$

Solution Steps

Combine 4x^{3} and -x^{3} to get 3x^{3}.

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

Combine -4x^{2} and x^{2} to get -3x^{2}.

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

Combine 4x and -x to get 3x.

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

Factor

$$\left(x-1\right)\left(x+4\right)\left(x^{2}+1\right)$$

Solution Steps

Multiply and combine like terms.

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

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

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

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

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

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.

$$x=-4$$

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

$$x^{2}+1=0$$

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 1 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{0±\sqrt{-4}}{2}$$

Polynomial x^{2}+1 is not factored since it does not have any rational roots.

$$x^{2}+1$$

Rewrite the factored expression using the obtained roots.

$$\left(x-1\right)\left(x+4\right)\left(x^{2}+1\right)$$