Solve 36+75r^2-21r^4

$$36 + 75 r ^ { 2 } - 21 r ^ { 4 }$$

Factor

$$3\left(-r-2\right)\left(r-2\right)\left(7r^{2}+3\right)$$

Solution Steps

Factor out 3.

$$3\left(12+25r^{2}-7r^{4}\right)$$

Consider 12+25r^{2}-7r^{4}. Find one factor of the form kr^{m}+n, where kr^{m} divides the monomial with the highest power -7r^{4} and n divides the constant factor 12. One such factor is 7r^{2}+3. Factor the polynomial by dividing it by this factor.

$$\left(7r^{2}+3\right)\left(-r^{2}+4\right)$$

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

$$\left(2-r\right)\left(2+r\right)$$

Reorder the terms.

$$\left(-r+2\right)\left(r+2\right)$$

Rewrite the complete factored expression. Polynomial 7r^{2}+3 is not factored since it does not have any rational roots.

$$3\left(7r^{2}+3\right)\left(-r+2\right)\left(r+2\right)$$

Evaluate

$$36+75r^{2}-21r^{4}$$