Problem 111 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The completely factorized form of the expression \((x+y)^{4}-100(x+y)^{2}\) is \((x+y)^{2}+10)(x+y+\sqrt{10})(x+y-\sqrt{10})\.

Step 1: Recognize the identity

Analyzing the given exercise, it's noticed to be a square difference: \(a^2 - b^2\), where \(a = (x + y)^{2}\) and \(b = 10\).

Step 2: Apply the square difference identity

Use the formula for square difference \(a^2 - b^2 = (a+b)(a-b)\) to factor the expression: \((x+y)^{4}-100(x+y)^{2} = ((x+y)^2 + 10)((x+y)^2 - 10)\)

Step 3: Factorize further

The second factor \((x+y)^2 - 10\) can be further factorized as \((x+y+\sqrt{10})(x+y-\sqrt{10})\). But first factor \((x + y)^2 + 10\) can not be factorized further.

Step 4: Write the final result

So, the completely factorized form of the given expression is \((x+y)^{2}+10)(x+y+\sqrt{10})(x+y-\sqrt{10})\.

Key Concepts

Difference of Squares

The concept of the "difference of squares" is a fundamental idea in algebra that makes it easier to factor certain types of polynomial expressions. Imagine the expression as a subtraction between squares of two terms: precisely, something like \(a^2 - b^2\). This expression can always be neatly factorized into two binomials, \((a - b)(a + b)\).

Let's unpack this with an example. Consider the expression \((x+y)^4 - 100(x+y)^2\) given in the exercise. To use the difference of squares, we need to identify both terms as squares. Here's the breakdown:

• First term, \(a = (x + y)^2\), eventually leads to the square \(a^2 = (x+y)^4\).
• Second term, \(b = 10\), has the square \(b^2 = 100\).

Recognizing this pattern allows us to transform the expression into \(((x+y)^2 + 10)((x+y)^2 - 10)\) thanks to the difference of squares identity.

Algebraic Identities

Algebraic identities simplify complex expressions and make factoring easier. One powerful identity is the difference of squares formula mentioned before. It's a specific type of algebraic identity that factors expressions neatly into two terms without needing more complex maneuvers.

In algebra, knowing these identities by heart – like \(a^2 - b^2 = (a - b)(a + b)\) – can save you time and effort.
These identities serve as shortcuts for factoring and solving equations, allowing you to recompose expressions quickly. When dealing with quadratics or higher-degree polynomials, spotting these can turn a daunting task into a straightforward one.

In our problem, identifying the expression \((x+y)^4 - 100(x+y)^2\) as a difference of squares facilitates the use of this identity:
The solution then becomes applying \((a - b)(a + b)\), making it simpler to handle, leading to the factors \(((x+y)^2 + 10)((x+y)^2 - 10)\). Such understanding and application are crucial in algebra.

Polynomial Expressions

Polynomial expressions are combinations of variables and coefficients using arithmetic operations like addition and multiplication. They are fundamental in algebra, representing quantities in equations. Polynomials can vary in complexity from simple linear forms to intricate higher-degree identities.

In our exercise, we approached polynomials nested inside one another: \((x+y)^2\) squared becomes \((x+y)^4\), and this formation leads to a second-degree expression in a broader context. Understanding how expressions like \((x+y)^4 - 100(x+y)^2\) work assists greatly when factoring is required.

• Understanding polynomials is key to learning how to rework them using algebraic techniques.
• Performing operations on polynomials, such as adding or subtracting, might change the degree or the coefficients. When factoring, identifying these changes is pivotal.

The task involves transforming a tangled polynomial into simpler factors. Learning how to interpret nested expressions lets us apply identities successfully, such as spotting differences of squares in higher-order polynomials.