Solve a^35/a+1=

$$\frac{ { a }^{ 35 } }{ a } +1=$$

Factor

$$\left(a^{2}+1\right)\left(a^{32}-a^{30}+a^{28}-a^{26}+a^{24}-a^{22}+a^{20}-a^{18}+a^{16}-a^{14}+a^{12}-a^{10}+a^{8}-a^{6}+a^{4}-a^{2}+1\right)$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 1 from 35 to get 34.

$$factor(a^{34}+1)$$

Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{34} and m divides the constant factor 1. One such factor is a^{2}+1. Factor the polynomial by dividing it by this factor. The following polynomials are not factored since they do not have any rational roots: a^{32}-a^{30}+a^{28}-a^{26}+a^{24}-a^{22}+a^{20}-a^{18}+a^{16}-a^{14}+a^{12}-a^{10}+a^{8}-a^{6}+a^{4}-a^{2}+1,a^{2}+1.

$$\left(a^{2}+1\right)\left(a^{32}-a^{30}+a^{28}-a^{26}+a^{24}-a^{22}+a^{20}-a^{18}+a^{16}-a^{14}+a^{12}-a^{10}+a^{8}-a^{6}+a^{4}-a^{2}+1\right)$$

Evaluate

$$a^{34}+1$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 1 from 35 to get 34.

$$a^{34}+1$$