Question

Divide and reduce. Try some by calculator. $$\frac{a c+a d+b c+b d}{c^{2}-d^{2}} \div(a+b)$$

Short answer

\(\frac{1}{c - d}\). After factoring and canceling common terms, the simplified result is \(\frac{1}{c - d}\).

Step-by-step solution

Factor the numerator

Factor by grouping the terms in the numerator. Group terms with common factors together: \(a c + a d + b c + b d = a(c + d) + b(c + d).\) Since both groups contain \(c + d \), factor it out: \(a(c + d) + b(c + d) = (a + b)(c + d).\)

Factor the denominator

Recognize the denominator as a difference of squares and factor it: \(c^2 - d^2 = (c + d)(c - d).\)

Setup the division

Set up the division of the fraction by \(a+b\): \[\frac{(a + b)(c + d)}{(c + d)(c - d)} \bigg/ (a+b) = \frac{(a + b)(c + d)}{(c + d)(c - d)} \times \frac{1}{a+b}.\]

Cancel common factors

Cancel out the common factors \(a+b\) and \(c+d\) from the numerator and denominator, including the ones from the division: \[\frac{(a + b)(c + d)}{(c + d)(c - d)} \times \frac{1}{a+b} = \frac{1}{c - d}.\] The remaining expression after canceling the common factors is the simplified result.

Key concepts

Difference of Squares

When we talk about the difference of squares in algebra, we're referring to expressions of the form \(a^2 - b^2\). This special pattern can be factored into \(a + b\) times \(a - b\). It's a fundamentally useful skill because it shows up often in algebra problems, including polynomial division, solving equations, and simplifying expressions.

For instance, take the expression \(c^2 - d^2\). It seems complicated at first glance, but knowing the difference of squares, we can simplify it to \(c + d\) times \(c - d\). This specific factorization is invaluable when simplifying complex fractions or when trying to cancel terms in a polynomial division. Recognizing this pattern rapidly makes more involved algebraic manipulations much easier to handle.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that together represent a specific value or set of values. These expressions can get very complex, and part of the challenge in algebra is simplifying them to a more manageable form.

In the given exercise, we see a compound algebraic expression in the numerator: \(a c + a d + b c + b d\). Instead of attempting to divide this messy expression as is, we 'factor by grouping' – a technique where we group terms with common variables and factor them out. By noticing that \(c+d\) appears with both \(a\) and \(b\), we can rewrite the expression as \(a(c + d) + b(c + d)\), and then factor out the common \(c+d\) to get \(a+b\) times \(c+d\). Doing this greatly simplifies the problem at hand.

Polynomial Division

Polynomial division is a way of dividing a polynomial by another polynomial, similar in concept to long division with numbers. One common scenario where polynomial division is used involves simplifying complex rational expressions.

In our exercise, once we have simplified the numerator and the denominator by factoring, we then set up the division of the entire fraction by \(a+b\). This division step looks daunting, but after simplification, it turns into a multiplication by the reciprocal of \(a+b\). After that, it becomes clear which terms cancel out, revealing the simplicity behind the seemingly complicated initial expression.

Simplifying Fractions

Simplifying fractions is a critical skill in algebra and entails reducing a complex fraction to its simplest form. We often look for common factors in the numerator and the denominator that can be canceled out.

In the provided exercise, after careful factoring and setting up the division, we are presented with a fraction that has a common \(a+b\) and \(c+d\) in the numerator and denominator. Cancelling out these common factors leads us to the simplified result \(\frac{1}{c - d}\). Simplifying fractions in such a manner is not just about making the expression shorter; it often reveals more about the underlying relationships between the algebraic terms involved.