Solve a^4-4a+3

$$a ^ { 4 } - 4 a + 3$$

Factor

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

Solution Steps

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. One such root is 1. Factor the polynomial by dividing it by a-1.

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

Consider a^{3}+a^{2}+a-3. 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. One such root is 1. Factor the polynomial by dividing it by a-1.

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

Rewrite the complete factored expression. Polynomial a^{2}+2a+3 is not factored since it does not have any rational roots.

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

Evaluate

$$a^{4}-4a+3$$