Problem 92 - Algebra A Combined Function

Question

Factor each expression completely. $$ m^{2}(n+8)-9(n+8) $$

Step-by-Step Solution

Answer

\((m+3)(m-3)(n+8)\)

Step 1: Identify the Common Factor

Observe the expression \( m^{2}(n+8)-9(n+8) \). Notice that the binomial \((n+8)\) is common in both terms. This suggests that you can use factoring by grouping.

Step 2: Factor by Grouping

Factor out the common binomial \((n+8)\) from both terms. This can be written as:\[(m^2 - 9)(n+8)\]

Step 3: Identify the Difference of Squares

Notice that the expression \(m^2 - 9\) is a difference of squares. The difference of squares formula is \(a^2 - b^2 = (a+b)(a-b)\).

Step 4: Apply the Difference of Squares Formula

Apply the formula to \(m^2 - 9\), where \(a = m\) and \(b = 3\). This gives:\[(m+3)(m-3)\]

Step 5: Complete the Factorization

Substitute \((m+3)(m-3)\) back into the expression, replacing \(m^2 - 9\):\[(m+3)(m-3)(n+8)\]

Step 6: Verify the Factorization

Ensure the factorization is correct by expanding \((m+3)(m-3)(n+8)\) back to the original expression:\((m^2 - 9)(n+8)\), which confirms the correct factorization.

Key Concepts

Difference of Squares

The concept of the difference of squares is essential in algebra, especially when it comes to factoring polynomials. At its core, it is a technique used to simplify expressions of the form \(a^2 - b^2\). The key characteristic of this expression is that it consists of two perfect squares separated by a subtraction sign. The formula for the difference of squares is:

• \(a^2 - b^2 = (a + b)(a - b)\)

To apply this in our example, notice that \(m^2 - 9\) fits perfectly into this pattern. Here, \(m^2\) is the square of \(m\) and \(9\) is the square of \(3\), making \(9 = 3^2\). By applying the difference of squares formula, we can rewrite the expression as:

• \((m + 3)(m - 3)\)

This simple technique elegantly reduces the complexity of such expressions and is a fundamental skill in algebra.

Factoring by Grouping

Factoring by grouping is a strategy used when expressions have terms that share common factors. It involves arranging terms into groups that can each be factored separately, and then combining these factors. This method works well for expressions where you can pinpoint similarities between the groups. Here's a step-by-step process that can help you understand it better:

• Identify terms that can be grouped: In the expression \(m^{2}(n+8) - 9(n+8)\), the binomial \((n+8)\) is common between the two terms.
• Factor the common term: Here, we take \((n+8)\) as the common factor out of the expression, leading to \((m^2 - 9)(n+8)\).

By focusing on the structure of the expression and recognizing common factors, you can transform and simplify complex expressions into more manageable forms.

Common Factor

Identifying a common factor is a crucial initial step in the factoring process. When working with polynomials, it becomes important to look for any term that is shared amongst different parts of the expression. The ability to spot and extract these common factors simplifies the expression significantly.

• Look for terms that occur in every part: In the expression \(m^{2}(n+8) - 9(n+8)\), \((n+8)\) appears in both terms.
• Factor out the common term to simplify: By taking out \((n+8)\) from both parts of the expression, we simplify it to \((m^2 - 9)(n+8)\).

This method not only simplifies the expression but also serves as a stepping stone in more complex operations. Recognizing common factors is foundational in the broader context of factoring polynomials, making it easier to handle even more intricate algebraic expressions.