Problem 50 - Elementary and Intermediate Algebra

Question

Find each special product. $$ \left(m^{2}-1\right)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \( m^4 - 2m^2 + 1 \).

Step 1: Recognize the Pattern

The expression \( \left(m^2 - 1\right)^2 \) is a perfect square trinomial which follows the pattern \( (a - b)^2 = a^2 - 2ab + b^2 \). Identify that \( a = m^2 \) and \( b = 1 \).

Step 2: Apply the Pattern

Apply the perfect square trinomial formula: \[ (m^2 - 1)^2 = (m^2)^2 - 2(m^2)(1) + 1^2 \] This expands to \( m^4 - 2m^2 + 1 \).

Step 3: Final Expression

The expanded form of the expression \( \left(m^2 - 1\right)^2 \) is now \( m^4 - 2m^2 + 1 \).

Key Concepts

Perfect Square Trinomials

To solve problems like \((m^2 - 1)^2\), recognizing the perfect square trinomial is key. A perfect square trinomial is an expression of the form \((a - b)^2\) or \((a + b)^2\). These patterns help simplify algebraic expressions quickly.
Here's how it works:

• \((a - b)^2 = a^2 - 2ab + b^2\)
• \((a + b)^2 = a^2 + 2ab + b^2\)

In our case, \((m^2 - 1)^2\) is identified as a perfect square trinomial. You identify this by setting \(a = m^2\) and \(b = 1\).
Recognizing these patterns allows you to apply the formula efficiently, streamlining algebraic expansions.

Polynomial Expansion

Expanding polynomials involves multiplying expressions to eliminate brackets. In our expression \((m^2 - 1)^2\), utilize the formula for a perfect square trinomial:
\[(a - b)^2 = a^2 - 2ab + b^2\]
For this example:

• First, calculate the square of \(a\): \((m^2)^2 = m^4\)
• Next, calculate \(-2ab\): \(-2\cdot m^2\cdot 1 = -2m^2\)
• Finally, calculate the square of \(b\): \(1^2 = 1\)

Now combine these results:
\(m^4 - 2m^2 + 1\)
This expanded form represents the polynomial without parentheses, simplifying its manipulation in further algebraic operations.

Algebraic Patterns

Recognizing algebraic patterns can drastically simplify problem-solving in algebra. These patterns are foundational shortcuts that help expand and factor expressions quickly.
In \((m^2 - 1)^2\), the algebraic pattern of a perfect square trinomial is applied. Algebraic patterns often resemble:

• Difference of squares: \((a^2 - b^2) = (a - b)(a + b)\)
• Perfect squares: \((a - b)^2\) or \((a + b)^2\)
• Cubic patterns: \((a - b)^3\) or \((a + b)^3\)

These patterns offer a structured approach to not only solving but also simplifying expressions. As you practice, identifying these will become second nature, making algebraic manipulation more intuitive and less error-prone.