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Chapter 6: Problem 50

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

Short Answer

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

Step by step solution

01

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 \( b = 1 \).
02

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 \( m^4 - 2m^2 + 1 \).
03

Final Expression

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Two students factor \(2 x^{2}+20 x+42\) and get two different answers: \((2 x+6)(x+7)\) and \((x+3)(2 x+14)\) Do both answers check? Why don't they agree? Is either answer completely correct? Explain.

Choose the correct method from Sections 6.1, 6.2, or 6.3 to factor each of the following. $$ x^{3}-2 x^{2}+5 x-10 $$

Choose the correct method from Sections 6.1, 6.2, or 6.3 to factor each of the following. $$ 9 x^{4}+27 x^{6} $$

Factor using rational numbers. $$ x(x-y)-y(y-x) $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 27 x-27 y-27 z $$
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