Question

Resolve \(\frac{a x+b^{2}}{x^{2}-(a+b)^{2}}\) into partial fractions. (1) \(\frac{a^{2}+a b+b^{2}}{2(a+b)(x-(a+b))}+\frac{a^{2}+a b-b^{2}}{2(a+b)(x+(a+b))}\) (2) \(\frac{a^{2}+b^{2}}{2(a+b)(x-(a+b))}+\frac{a^{2}-b^{2}}{2(a+b)(x+(a+b))}\) (3) \(\frac{a^{2}-b^{2}}{2(a+b)(x-(a-b))}+\frac{a^{2}+b^{2}}{2(a+b)(x+(a+b))}\) (4) None of these

Short answer

Now, write a short answer: The given function, $\frac{a x+b^{2}}{x^{2}-(a+b)^{2}}$, can be written as the sum of partial fractions: $\frac{a^2+ab+b^2}{2(a+b)(x-(a+b))}+ \frac{a^2-ab+b^2}{2(a+b)(x+(a+b))}$. The correct option is the first one.

Step-by-step solution

Identify the form of the function

The given function is \(\frac{a x+b^{2}}{x^{2}-(a+b)^{2}}\). The denominator is a difference of squares.

Factor the denominator

Rewrite the denominator as a product of factors: \(x^2 - (a+b)^2 = (x-(a+b))(x+(a+b))\).

Use partial fractions

Now, we'll break the given fraction into partial fractions: $$ \frac{a x+b^{2}}{(x-(a+b))(x+(a+b))} = \frac{A}{x-(a+b)}+ \frac{B}{x+(a+b)} $$ We will solve for the coefficients A and B.

Solve for coefficients A and B

Multiply both sides by the common denominator and clear the fractions: \( a x+b^{2} = A(x+(a+b)) + B(x-(a+b))\) Now, we'll solve for A and B by substitution method.

Substituting the value of x

We can solve for the values of A and B by selecting appropriate values for x, which will eliminate each term separately. Let x = a + b, then we get: \(a(a+b)+b^2 = A(2(a+b))\) Now, divide by \(2(a+b)\): \(A = \frac{a^2+ab+b^2}{2(a+b)}\) For B, let x = -(a+b), then we get: \(-a(a+b)+b^2 = B(-2(a+b))\) Now, divide by \(-2(a+b)\): \(B = \frac{a^2-ab+b^2}{2(a+b)}\)

Putting it all together

Now, replace A and B with the calculated values to write the given function as a sum of partial fractions: $$ \frac{a x+b^{2}}{(x-(a+b))(x+(a+b))} = \frac{a^2+ab+b^2}{2(a+b)(x-(a+b))}+ \frac{a^2-ab+b^2}{2(a+b)(x+(a+b))} $$ Comparing our result to the options provided, we find that our solution matches the first option: \(\frac{a^{2}+a b+b^{2}}{2(a+b)(x-(a+b))}+\frac{a^{2}+a b-b^{2}}{2(a+b)(x+(a+b))}\).

Key concepts

Factorization

Factorization is a fundamental skill in algebra, particularly useful when dealing with polynomials and rational expressions. In this context, it involves breaking down the denominator of a rational expression into simpler parts, which can then be used to simplify the expression or solve equations.

The original problem presents the denominator as a difference of squares: \(x^2 - (a+b)^2\). The difference of squares is a special algebraic identity:

• a² - b² = (a - b)(a + b)

This identity allows us to rewrite the expression into a product of two binomials. In this exercise, the denominator \(x^2 - (a+b)^2\) factors into \((x-(a+b))(x+(a+b))\).

Understanding these identities is essential, as they simplify complex expressions and make equations more manageable. Without factorization, working with rational expressions like the one in the problem would be much more difficult.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In the context of solving rational expressions through partial fractions, algebra allows us to break down complex expressions into simpler components.

The exercise involves rewriting a rational expression as a sum of simpler fractions, each having linear denominators. The given expression is:

• \(\frac{a x+b^{2}}{(x-(a+b))(x+(a+b))}\)

In algebra, the goal of partial fraction decomposition is to express this as:

• \(\frac{A}{x-(a+b)} + \frac{B}{x+(a+b)}\)

The process requires finding constants \(A\) and \(B\) which satisfy the equation when both sides are multiplied by the original denominator, eliminating the fractions.

The substitution method helps determine these constants by substituting specific values of \(x\), eliminating terms, and solving for \(A\) and \(B\). Algebraically solving these can often involve multiple substitution steps to simplify and find the correct coefficients, as demonstrated in the problem solution.

Rational Expressions

Rational expressions are fractions in which the numerator and/or the denominator are polynomials. They are fundamental in algebra and calculus. Understanding how to simplify, add, subtract, multiply, and divide rational expressions is crucial for higher mathematics.

In the exercise, we're looking at simplifying a rational expression by breaking it down into partial fractions. The expression \(\frac{a x+b^{2}}{x^{2}-(a+b)^{2}}\) needs to be decomposed into a simpler form with linear denominators.

• The goal is to write it as a sum of fractions, each with a simpler denominator.
• Partial fraction decomposition is a technique that assists in integrating rational expressions or simplifying their form for further analysis.

By rewriting the original rational expression via partial fraction decomposition, we make it easier to work with in various algebraic contexts, like integration or transformation. Breaking down complex rational expressions into simpler parts is an invaluable skill that often appears in both academic and practical applications.