Question

Find a conjugate of each expression and the product of the expression with the conjugate. $$ 1-\sqrt{r+1} $$

Short answer

Question: Find the product of the expression \(1-\sqrt{r+1}\) and its conjugate. Answer: \(-r\)

Step-by-step solution

Identify the expression and its terms

The given expression is \(1-\sqrt{r+1}\). We can see that this is a binomial expression with two terms: 1 and \(-\sqrt{r+1}\).

Find the conjugate of the expression

The conjugate of the given expression can be found by changing the sign of the second term. Thus, the conjugate of \(1-\sqrt{r+1}\) will be \(1+\sqrt{r+1}\).

Find the product of the expression and its conjugate

Now, we will multiply the given expression by its conjugate to eliminate the square root term and simplify the expression: $$(1-\sqrt{r+1})(1+\sqrt{r+1})$$ To find the product, we use the FOIL method (first, outer, inner, last): $$1(1)+1(\sqrt{r+1})-\sqrt{r+1}(1)-\sqrt{r+1}(\sqrt{r+1})$$ Combine the terms and simplify the expression: $$1+\sqrt{r+1}-\sqrt{r+1}-(r+1)$$ Now, the positive and negative \(\sqrt{r+1}\) terms cancel out, $$1-(r+1)$$ Finally, simplify the expression by distributing the negative sign: $$1-r-1$$ The final product of the expression and its conjugate is \(-r\).

Key concepts

Binomial Expression

The term "binomial expression" refers to an algebraic expression that contains exactly two terms. It's like a duo working together under one roof. In our example, the binomial expression is \(1 - \sqrt{r+1}\). Here, the two terms are \(1\) and \(-\sqrt{r+1}\). Recognizing an expression as a binomial is key in algebra, especially when you need to find its conjugate.
A conjugate is a special kind of expression where we change the sign between two terms in a binomial. By finding the conjugate, we can perform operations such as multiplying the expression by its conjugate to simplify or eliminate certain components, such as square roots. This technique is handy when we aim to rationalize an expression and tidy up messy radicals.

Square Roots

Square roots are quite common in math, and they show up whenever we want to find a number which, when multiplied by itself, gives another number. In our binomial expression \(1 - \sqrt{r+1}\), \(\sqrt{r+1}\) is the square root term.
Simplifying expressions involving square roots often requires some nifty math tricks, especially eliminating the square roots to make calculations simpler. By multiplying a number by the square root of itself, we reach a base level, which is the original "non-square-root" number, such as when \(\sqrt{r+1} \times \sqrt{r+1} = r+1\).

• This process is crucial for the simplification of equations and understanding higher-level math concepts.
• Square roots can be challenging at times, but with practice, they become easier to manage.

Simplification

Simplification in mathematics involves making equations or expressions easier to handle or understand. During the process of multiplying a binomial expression with its conjugate, one of our goals is to simplify the expression.
When we perform the multiplication \((1 - \sqrt{r+1})(1 + \sqrt{r+1})\), we see the square root terms \(+\sqrt{r+1}\) and \(-\sqrt{r+1}\) cancel each other out. This process reduces tedious complexity and leads to a simplified equation: \(1 - (r+1)\).
Here's a general guideline for simplification:

• Eliminate terms that cancel each other out.
• Combine like terms.
• Distribute constants across terms effectively.

This clean-up process ensures our final answer is presented in the simplest form, making problems easier to solve and understand.

FOIL Method

The FOIL method is a technique used to multiply two binomials. It's a simple mnemonic to remember: First, Outer, Inner, Last. Let's break it down:

• **First**: Multiply the first terms in each binomial.
• **Outer**: Multiply the outer terms in the product.
• **Inner**: Multiply the inner terms.
• **Last**: Multiply the last terms in each binomial.

In the case of our expression \((1 - \sqrt{r+1})(1 + \sqrt{r+1})\), applying FOIL looks like this:

• First: \(1 \cdot 1 = 1\)
• Outer: \(1 \cdot \sqrt{r+1} = \sqrt{r+1}\)
• Inner: \(-\sqrt{r+1} \cdot 1 = -\sqrt{r+1}\)
• Last: \(-\sqrt{r+1} \cdot \sqrt{r+1} = -(r+1)\)

When done correctly, the middle terms, \(+\sqrt{r+1}\) and \(-\sqrt{r+1}\), cancel each other out, simplifying the expression dramatically. The FOIL method is powerful for handling polynomial expressions, providing a systematic way to break down and solve products of binomials.