Solve x^4+3x^3-x-3.x^3+x^2-5x+3

$$x ^ { 4 } + 3 x ^ { 3 } - x - 3 . x ^ { 3 } + x ^ { 2 } - 5 x + 3$$

Factor

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

Solution Steps

Do the grouping x^{4}+3x^{3}-x-3=\left(x^{4}+3x^{3}\right)+\left(-x-3\right), and factor out x^{3} in the first and -1 in the second group.

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

Factor out common term x+3 by using distributive property.

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

Consider x^{3}-1. Rewrite x^{3}-1 as x^{3}-1^{3}. The difference of cubes can be factored using the rule: a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right).

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

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

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

Evaluate

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