Problem 112 - Algebra and Trigonometry

Question

Factor completely. $$(x+y)^{4}-100(x+y)^{2}$$

Step-by-Step Solution

Answer

So, the factored form of the given expression \( (x+y)^{4}-100(x+y)^{2} \) is \( [(x+y)^{2} - 10][(x+y)^{2} + 10] \).

Step 1 - Identify the Form

Recognize the given expression is in the form of a quadratic expression \( a^2 - b^2 \), which is a difference of two squares.

Step 2 - Apply the Difference of Squares Formula

Apply the difference of squares formula, \( a^2 - b^2 = (a-b)(a+b) \), to our expression. In this case, \( a = (x+y)^2 \) and \( b = 10 \) giving us \( [(x+y)^{2} - 10][(x+y)^{2} + 10] \) as an initial factoring.

Step 3 - Factor Further if Possible

Examine each factor to see if it can be factored further. The expression \( (x+y)^2 - 10 \) looks like a binomial that cannot be factored further. However, \( (x+y)^2 \) could be written as \( (x+y)(x+y) \) – but since it does not simplify the expression further, we choose to leave it in its original form.