Solve t^24-t^4

$${ t }^{ 24 } - { t }^{ 4 }$$

Factor

$$\left(t-1\right)\left(t+1\right)\left(t^{2}+1\right)t^{4}\left(t^{4}-t^{3}+t^{2}-t+1\right)\left(t^{4}+t^{3}+t^{2}+t+1\right)\left(t^{8}-t^{6}+t^{4}-t^{2}+1\right)$$

Solution Steps

Factor out t^{4}.

$$t^{4}\left(t^{20}-1\right)$$

Consider t^{20}-1. Rewrite t^{20}-1 as \left(t^{10}\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(t^{10}-1\right)\left(t^{10}+1\right)$$

Consider t^{10}-1. Rewrite t^{10}-1 as \left(t^{5}\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(t^{5}-1\right)\left(t^{5}+1\right)$$

Consider t^{5}-1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 1. One such root is 1. Factor the polynomial by dividing it by t-1.

$$\left(t-1\right)\left(t^{4}+t^{3}+t^{2}+t+1\right)$$

Consider t^{5}+1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient 1. One such root is -1. Factor the polynomial by dividing it by t+1.

$$\left(t+1\right)\left(t^{4}-t^{3}+t^{2}-t+1\right)$$

Consider t^{10}+1. Find one factor of the form t^{k}+m, where t^{k} divides the monomial with the highest power t^{10} and m divides the constant factor 1. One such factor is t^{2}+1. Factor the polynomial by dividing it by this factor.

$$\left(t^{2}+1\right)\left(t^{8}-t^{6}+t^{4}-t^{2}+1\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: t^{4}-t^{3}+t^{2}-t+1,t^{4}+t^{3}+t^{2}+t+1,t^{8}-t^{6}+t^{4}-t^{2}+1,t^{2}+1.

$$t^{4}\left(t-1\right)\left(t^{4}+t^{3}+t^{2}+t+1\right)\left(t+1\right)\left(t^{4}-t^{3}+t^{2}-t+1\right)\left(t^{2}+1\right)\left(t^{8}-t^{6}+t^{4}-t^{2}+1\right)$$

Evaluate

$$t^{4}\left(t^{20}-1\right)$$