Solve a^3-y^3/27-a^2y+ay^2/3+1

$$a ^ { 3 } - \frac { y ^ { 3 } } { 27 } - a ^ { 2 } y + \frac { a y ^ { 2 } } { 3 } + 1$$

Factor

$$\frac{\left(3+3a-y\right)\left(y^{2}-6ay+3y+9a^{2}-9a+9\right)}{27}$$

Solution Steps

Factor out \frac{1}{27}.

$$\frac{27a^{3}-y^{3}-27a^{2}y+9ay^{2}+27}{27}$$

Consider 27a^{3}-y^{3}-27a^{2}y+9ay^{2}+27. Consider 27a^{3}-y^{3}-27a^{2}y+9ay^{2}+27 as a polynomial over variable a.

$$27a^{3}-27ya^{2}+9y^{2}a-y^{3}+27$$

Find one factor of the form ka^{m}+n, where ka^{m} divides the monomial with the highest power 27a^{3} and n divides the constant factor -y^{3}+27. One such factor is -y+3a+3. Factor the polynomial by dividing it by this factor.

$$\left(-y+3a+3\right)\left(y^{2}-6ay+3y+9a^{2}-9a+9\right)$$

Rewrite the complete factored expression.

$$\frac{\left(-y+3a+3\right)\left(y^{2}-6ay+3y+9a^{2}-9a+9\right)}{27}$$

Evaluate

$$\frac{\left(3+3a-y\right)\left(\left(3a-y\right)^{2}+3y-9a+9\right)}{27}$$