Questions: Factor. Assume that variables in exponents represent natural numbers. 56 y^4 - 117 x y^2 + 36 x^2 The complete factorization is (Factor completely. Simplify your answer.)

Solution Steps

To factor the given polynomial \(56 y^{4} - 117 x y^{2} + 36 x^{2}\), we can treat it as a quadratic in terms of \(y^2\). We will look for two binomials that multiply to give the original polynomial.

1. Recognize the polynomial as a quadratic in terms of \(y^2\).
2. Use the quadratic formula or factoring techniques to find the roots.
3. Express the polynomial as a product of binomials.

We start with the polynomial given in the problem: \[ 56 y^{4} - 117 x y^{2} + 36 x^{2} \] This can be rearranged as: \[ 36 x^{2} - 117 x y^{2} + 56 y^{4} \]

Next, we factor the polynomial. We can express it as a product of two binomials: \[ (3x - 8y^{2})(12x - 7y^{2}) \]

Thus, the complete factorization of the polynomial \(56 y^{4} - 117 x y^{2} + 36 x^{2}\) is: \[ (3x - 8y^{2})(12x - 7y^{2}) \]

Final Answer