Factor 16x^4-81y^4

Problem

16*x^4−81*y^4

Solution

1. Identify the expression as a difference of squares, which follows the pattern a^2−b^2=(a−b)*(a+b)

2. Rewrite the terms as perfect squares.

16*x^4=(4*x^2)^2

81*y^4=(9*y^2)^2

3. Apply the formula for the difference of squares to the expression.

16*x^4−81*y^4=(4*x^2−9*y^2)*(4*x^2+9*y^2)

4. Identify that the first factor (4*x^2−9*y^2) is also a difference of squares.

4*x^2=(2*x)^2

9*y^2=(3*y)^2

5. Factor the term (4*x^2−9*y^2) using the difference of squares formula again.

4*x^2−9*y^2=(2*x−3*y)*(2*x+3*y)

6. Combine all factors into the final expression. Note that (4*x^2+9*y^2) is a sum of squares and cannot be factored further using real coefficients.

Final Answer

16*x^4−81*y^4=(2*x−3*y)*(2*x+3*y)*(4*x^2+9*y^2)

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