Solve 8by^{3}+12b^2y^4+4b^3y^5

$$8 b y ^ { 3 } + 12 b ^ { 2 } y ^ { 4 } + 4 b ^ { 3 } y ^ { 5 }$$

Factor

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

Solution Steps

Factor out 4.

$$4\left(2by^{3}+3b^{2}y^{4}+b^{3}y^{5}\right)$$

Consider 2by^{3}+3b^{2}y^{4}+b^{3}y^{5}. Factor out by^{3}.

$$by^{3}\left(2+3by+b^{2}y^{2}\right)$$

Consider 2+3by+b^{2}y^{2}. Consider 2+3by+b^{2}y^{2} as a polynomial over variable b.

$$y^{2}b^{2}+3yb+2$$

Find one factor of the form y^{k}b^{m}+n, where y^{k}b^{m} divides the monomial with the highest power y^{2}b^{2} and n divides the constant factor 2. One such factor is by+2. Factor the polynomial by dividing it by this factor.

$$\left(by+2\right)\left(by+1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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