Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\((m-3+n)(m-3-n)\)

Step-by-Step Solution

Answer

The product of \((m-3+n)(m-3-n)\) is \(m^2 - 6m + 9 - n^2\).

Step 1: Understand the Product of Conjugates Rule

The product of conjugates rule states that \((a+b)(a-b) = a^2 - b^2\). This means when multiplying two expressions that have the same terms but opposite signs in the middle, only the first term (squared) and the second term (squared) matter, and they're subtracted from each other.

Step 2: Apply the Rule to the Exercise

Applying this rule to the given exercise, here \(a=m-3\) and \(b=n\). So, the product will be \((m-3)^2 - n^2\).

Step 3: Calculate the Square Terms

To further simplify the expression, compute the square of \(m-3\) and \(n\). This results in \(m^2 - 6m + 9 - n^2\).

Key Concepts

Product of Conjugates

In algebra, understanding the product of conjugates is a handy tool that simplifies expressions involving two binomials with differing signs between the terms. The product of conjugates rule is straightforward:

• If you have \((a + b)(a - b)\), the result is simply \(a^2 - b^2\).
• This rule applies because the middle terms of the binomials cancel out, leaving behind the squared differences.

For the expression \((m-3+n)(m-3-n)\), we identify \(a = m - 3\) and \(b = n\), thus simplifying it using this rule gives us the expression \((m-3)^2 - n^2\). This eliminates the need to expand the entire multiplication, simplifying the solution significantly. The product of conjugates is a quick route to find differences of squares, saving time and effort in polynomial multiplication.

Polynomial Simplification

Polynomial simplification involves reducing expressions to their simplest form. In this context, after applying the product of conjugates rule, the expression simplifies to \((m-3)^2 - n^2\). The next step is to break down these squares.

• Calculate \((m-3)^2\), which involves expanding \((m-3)(m-3)\) into \(m^2 - 6m + 9\).
• The term \(-n^2\) remains unchanged in this step.

Putting it all together, we arrive at the simplified expression: \(m^2 - 6m + 9 - n^2\).
This simplification is essential because it provides a clearer view of the polynomial, showing how terms combine and helping in further algebraic manipulations or solving equations involving these terms.

Binomial Theorem

Though not directly used in the original problem, understanding the Binomial Theorem can provide context and methods for expanding binomials. The Binomial Theorem states that \((a + b)^n\) can be expanded as a sum involving terms \(\binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) are the binomial coefficients. This formula gives a framework for expanding expressions raised to a power without multiplying out each term individually.
For instance, if you are tasked with expanding \((m-3+n)^2\) using the Binomial Theorem, identify \(a = m-3\) and \(b = n\), then rewrite it as \((m-3)^2 + 2(m-3)n + n^2\). This helps see how patterns in expansions emerge. While this theorem wasn't necessary for the original solution of the product of conjugates, it's crucial for understanding and working with higher powers and more complex polynomial expressions in algebra.