Solve 4x^8-61x^4+225

$$4 x ^ { 8 } - 61 x ^ { 4 } + 225$$

Factor

$$\left(2x^{2}-5\right)\left(x^{2}-3\right)\left(2x^{2}+5\right)\left(x^{2}+3\right)$$

Solution Steps

Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 4x^{8} and n divides the constant factor 225. One such factor is 4x^{4}-25. Factor the polynomial by dividing it by this factor.

$$\left(4x^{4}-25\right)\left(x^{4}-9\right)$$

Consider 4x^{4}-25. Rewrite 4x^{4}-25 as \left(2x^{2}\right)^{2}-5^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2x^{2}-5\right)\left(2x^{2}+5\right)$$

Consider x^{4}-9. Rewrite x^{4}-9 as \left(x^{2}\right)^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

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

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: 2x^{2}-5,x^{2}-3,x^{2}+3,2x^{2}+5.

$$\left(2x^{2}-5\right)\left(x^{2}-3\right)\left(x^{2}+3\right)\left(2x^{2}+5\right)$$

Evaluate

$$4x^{8}-61x^{4}+225$$