Solve 41a^{3}b^{3}+34a^2b^2+12a^4b^4

$$41 a ^ { 3 } b ^ { 3 } + 34 a ^ { 2 } b ^ { 2 } + 12 a ^ { 4 } b ^ { 4 }$$

Factor

$$a^{2}b^{2}\left(12ab+17\right)\left(ab+2\right)$$

Solution Steps

Factor out a^{2}b^{2}.

$$a^{2}b^{2}\left(41ab+34+12a^{2}b^{2}\right)$$

Consider 41ab+34+12a^{2}b^{2}. Consider 41ab+34+12a^{2}b^{2} as a polynomial over variable a.

$$12b^{2}a^{2}+41ba+34$$

Find one factor of the form kb^{m}a^{n}+p, where kb^{m}a^{n} divides the monomial with the highest power 12b^{2}a^{2} and p divides the constant factor 34. One such factor is 12ab+17. Factor the polynomial by dividing it by this factor.

$$\left(12ab+17\right)\left(ab+2\right)$$

Rewrite the complete factored expression.

$$a^{2}b^{2}\left(12ab+17\right)\left(ab+2\right)$$

Evaluate

$$\left(12ab+17\right)\left(ab+2\right)\left(ab\right)^{2}$$