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Chapter 5: Problem 59

Divide as indicated. $$\frac{x^{4}+y^{4}}{x+y}$$

Short Answer

Expert verified
The simplified form of the expression is \(x - y\sqrt{2}\) and \(x + y\sqrt{2}\).

Step by step solution

01

Factorize the numerator

Write the expression as \(x^{4}+y^{4} = (x^{2} + y^{2} +xy\sqrt{2} )*(x^{2} + y^{2} - xy\sqrt{2})\)
02

Break Down the Division

Now, divide each factorized part by \(x+y\). Hence, the expression will be \(\frac{x^{2} + y^{2} +xy\sqrt{2}}{x+y}\) and \(\frac{x^{2} + y^{2} - xy\sqrt{2}}{x+y}\)
03

Solving the Simplified Expressions

Solve for each expression. The first term simplifies to \(x - y\sqrt{2}\), and the second term simplifies to \(x + y\sqrt{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Algebraic Expressions
Algebraic expressions are a combination of variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. They are the foundation of algebra and help us describe mathematical relationships. For example, in the expression \(x^4 + y^4\), \(x\) and \(y\) are variables, while 4 is an exponent.
Understanding how to manipulate these expressions, such as through division, is crucial in algebra. When you're dividing expressions, you aim to simplify them into more manageable parts. This often involves operations like factoring and then canceling common terms.
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of simpler polynomials. This technique breaks down higher-degree terms into factors that are easier to manipulate and divide. In any polynomial like \(x^4 + y^4\), factoring involves recognizing patterns or using formulas to rewrite it.
For example, in our given problem, the expression \(x^4 + y^4\) is factored into two parts: \((x^2 + y^2 + xy\sqrt{2})(x^2 + y^2 - xy\sqrt{2})\). This step is essential because it simplifies the division between the numerator and the denominator.
Simplifying Expressions
Simplifying expressions is an essential skill that makes solving algebraic equations more straightforward. Once we factorize the polynomial, dividing it by \(x+y\) becomes easier.
In our example, we have divided both factors by \(x+y\):
  • \(\frac{x^2 + y^2 + xy\sqrt{2}}{x+y} = x - y\sqrt{2}\)
  • \(\frac{x^2 + y^2 - xy\sqrt{2}}{x+y} = x + y\sqrt{2}\)
Simplifying transforms complex expressions into simpler forms like \(x - y\sqrt{2}\) and \(x + y\sqrt{2}\), which are much easier to work with in further mathematical computations.

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Most popular questions from this chapter

What polynomial, when divided by \(3 x^{2},\) yields the trinomial \(-6 x^{6}-9 x^{4}+12 x^{2}\) as a quotient?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$5 x^{2} \cdot 4 x^{6}=9 x^{8}$$

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\left(2 x^{2}-8 x+6\right)-\left(x^{2}-3 x+5\right)=x^{2}-5 x+1\) for any value of \(x .\)

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