Solve 4y^{6}-4xy^{5}+17x^{2}y^{4}-28x^{3}y^{3}+25x^6-40x^5y+26x^4y^2

$$4 y ^ { 6 } - 4 x y ^ { 5 } + 17 x ^ { 2 } y ^ { 4 } - 28 x ^ { 3 } y ^ { 3 } + 25 x ^ { 6 } - 40 x ^ { 5 } y + 26 x ^ { 4 } y ^ { 2 }$$

Factor

$$\left(y-x\right)^{2}\left(25x^{4}+21x^{2}y^{2}+4xy^{3}+4y^{4}+10yx^{3}\right)$$

Solution Steps

Consider 4y^{6}-4xy^{5}+17x^{2}y^{4}-28x^{3}y^{3}+25x^{6}-40x^{5}y+26x^{4}y^{2} as a polynomial over variable y.

$$4y^{6}-4xy^{5}+17x^{2}y^{4}-28x^{3}y^{3}+26x^{4}y^{2}-40x^{5}y+25x^{6}$$

Find one factor of the form ky^{m}+n, where ky^{m} divides the monomial with the highest power 4y^{6} and n divides the constant factor 25x^{6}. One such factor is -x+y. Factor the polynomial by dividing it by this factor.

$$\left(-x+y\right)\left(-25x^{5}+17x^{2}y^{3}+4y^{5}-11y^{2}x^{3}+15yx^{4}\right)$$

Consider -25x^{5}+17x^{2}y^{3}+4y^{5}-11y^{2}x^{3}+15yx^{4}. Consider -25x^{5}+17x^{2}y^{3}+4y^{5}-11y^{2}x^{3}+15yx^{4} as a polynomial over variable y.

$$4y^{5}+17x^{2}y^{3}-11x^{3}y^{2}+15x^{4}y-25x^{5}$$

Find one factor of the form py^{q}+u, where py^{q} divides the monomial with the highest power 4y^{5} and u divides the constant factor -25x^{5}. One such factor is -x+y. Factor the polynomial by dividing it by this factor.

$$\left(-x+y\right)\left(25x^{4}+21x^{2}y^{2}+4xy^{3}+4y^{4}+10yx^{3}\right)$$

Rewrite the complete factored expression.

$$\left(25x^{4}+21x^{2}y^{2}+4xy^{3}+4y^{4}+10yx^{3}\right)\left(-x+y\right)^{2}$$

Evaluate

$$\left(y-x\right)^{2}\left(25x^{4}+4xy^{3}+4y^{4}+10yx^{3}+21\left(xy\right)^{2}\right)$$