Solve 9+21r^2+12r^4

$$9 + 21 r ^ { 2 } + 12 r ^ { 4 }$$

Factor

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

Solution Steps

Factor out 3.

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

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

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

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

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

Evaluate

$$12r^{4}+21r^{2}+9$$