Question

For \(n \in N_{1} a^{2 n-1}+b^{2 n-1}\) is divisible by (1) \(a+b\) (2) \((a+b)^{2}\) (3) \(a^{3}+b^{3}\) (4) \(a^{2}+b^{2}\)

Short answer

Answer: (1) (a+b)

Step-by-step solution

Express the given expression in terms of sum of exponents

We are given the expression \(a^{2n-1} + b^{2n-1}\). Let's rewrite this expression in terms of the sum of exponents using the binomial theorem: \((a+b)\left[a^{2n-2}-a^{2n-3}b^{1}+a^{2n-4}b^{2}-\cdots\right]\) Now let's investigate the divisibility of the given expression by each of the given options.

Check the divisibility by \((a+b)\)

It's clear that the expression we derived in step 1 has \((a+b)\) as a factor. So the given expression is divisible by \((a+b)\). Hence, option (1) is true.

Check the divisibility by \((a+b)^{2}\)

Let's check if the given expression is divisible by \((a+b)^{2}\): We can rewrite \((a+b)^{2}\) as \(a^2 + 2ab + b^2\). Now, let's try to divide the original expression by \((a+b)^{2}\). If we perform this division, it becomes clear that the expression will not be divisible by \((a+b)^{2}\). Thus, option (2) is false.

Check the divisibility by \(a^{3}+b^{3}\)

We will use the identity \(a^{3} + b^{3} = (a+b)(a^2 - ab + b^2)\). We can see that if we try to divide the original expression by \(a^{3} + b^{3}\), it will not be an exact divisor. Therefore, option (3) is false.

Check the divisibility by \(a^{2} + b^{2}\)

Now let's check if the given expression is divisible by \(a^2 + b^2\). If we try to divide the expression by \(a^2 + b^2\), it will not be an exact divisor. Therefore, option (4) is false.

Conclusion

Based on our analysis, we can conclude that the expression \(a^{2n-1} + b^{2n-1}\) is divisible by \((a+b)\) (Option 1). The other options are not divisible by the given expression.

Key concepts

Sum of Exponents

Exponents represent numbers raised to a specific power. In the expression \(a^{2n-1} + b^{2n-1}\), each term showcases the sum of exponents. Understanding this helps in breaking down complex expressions into factorable parts. Here, the exponents in each term reflect a relationship between \(a\) and \(b\), making it suitable for certain divisibility tests.
By representing both \(a\) and \(b\) raised to similar powers, we can further manipulate the equation with various algebraic methods. Recognizing the pattern in the exponents is crucial. This recognition allows us to apply mathematical identities like the Binomial Theorem efficiently later on.

Binomial Theorem

The Binomial Theorem is a foundational tool in algebra for expanding powers of binomials. It allows us to express \((a + b)^n\) as a sum of terms where each term includes powers of \(a\) and \(b\).
In our scenario, the Binomial Theorem provides a structured way to explore the expression \(a^{2n-1} + b^{2n-1}\) by looking at its components, such as \((a+b)\) in the expression.

• Each term in the expansion has a coefficient that corresponds to binomial coefficients.
• This enables us to understand how higher powers of expressions, like \(a^n\), can still be factored or manipulated algebraically.

Algebraic Expression Factorization

Factorization is critical in simplifying expressions and solving equations. For \(a^{2n-1} + b^{2n-1}\), factorization involves expressing the equation in terms of simpler products, namely \((a+b)\).
This process begins by recognizing common factors or patterns in terms, and then extracting them.
Factorization can utilize identities and understanding of algebraic principles to break down seemingly complex expressions. This is evident when identifying that \((a+b)\) is a factor in our expression, allowing for an easy divisibility check.

Divisibility Tests

Divisibility tests are essential tools for verifying if one number can be divided by another without a remainder. In algebra, this involves factorization and knowledge of identities. With \(a^{2n-1} + b^{2n-1}\), divisibility tests help determine if expressions like \((a+b)\) are viable divisors.
By deriving the expression \((a+b)[...]\), we confirmed divisibility by \((a+b)\). Other expressions like \((a+b)^2\) or \(a^3 + b^3\) require additional identities to test divisibility but don't satisfy the condition for exact division without a remainder.
These tests highlight how algebraic manipulation and factorization are intertwined in divisibility analysis.

Mathematical Identities

Mathematical identities are equations true for all values of the variables within them. They are keys to simplifying and solving algebraic expressions. In our problem, identities like \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\) enable us to simplify and check divisibility.
Using known identities, we dissect \(a^{2n-1} + b^{2n-1}\) into manageable parts, revealing potential factors such as \((a+b)\). This step is crucial for concluding that \(a+b\) is a divisor.

• Identities provide shortcut methods to expand and factor expressions.
• They play a vital role in algebra by transforming complex problems into simpler tasks.
• In this problem, reliance on identities justified our conclusion about divisibility.