Problem 94 - Algebra A Combined Function

Question

Fill in the squares so that a true statement forms. $$ \left(5 x^{\square}-2\right)^{2}=25 x^{6}-20 x^{3}+4 $$

Step-by-Step Solution

Answer

Fill the square with 3: \((5x^3 - 2)^2 = 25x^6 - 20x^3 + 4\).

Step 1: Understand the Expression

The given expression is \((5x^{\square} - 2)^2\) on the left side and it needs to match the expanded form \(25x^6 - 20x^3 + 4\) on the right side. We need to find the value of \( \square \) to make these expressions equal.

Step 2: Recognize the Perfect Square Trinomial

The expression on the right side, \(25x^6 - 20x^3 + 4\), is in the form \((a - b)^2 = a^2 - 2ab + b^2\). We can identify \(a^2 = 25x^6\), \(2ab = 20x^3\), and \(b^2 = 4\). Determine \(a\) and \(b\).

Step 3: Determine Each Term

Since \(a^2 = 25x^6\), we find \(a = 5x^3\). And from \(b^2 = 4\), we have \(b = 2\). This matches \(\pm 2\), but since it fits into a binomial, we'll use \(-2\).

Step 4: Verify with the Middle Term

For the middle term \(2ab = 20x^3\), substituting \(a = 5x^3\) and \(b = 2\), we calculate: \[ 2ab = 2 \cdot 5x^3 \cdot 2 = 20x^3. \]This confirms our assumptions of values for \(a\) and \(b\).

Step 5: Identify the Power of x

In the original equation \((5x^{\square} - 2)\), we have \(5x^{\square} = 5x^3\), leading us to \(x^{\square} = x^3\). Thus, the power is \(3\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special kind of polynomial. It results from the expansion of a binomial squared. For instance, when you square a binomial of the form \((a - b)^2\), the result is \(a^2 - 2ab + b^2\).

In the original problem, the expression \((5x^{\square} - 2)^2\) illustrates this concept. The expanded form, given as \(25x^6 - 20x^3 + 4\), follows the perfect square trinomial pattern:

• \(a^2\) equals \(25x^6\), which implies that \(a\) must be \(5x^3\).
  
• \(b^2\) equals \(4\), leading to \(b\) being \(2\) (considering the positive constant in the context).
  
• The middle term \(-2ab\) is checked as \(20x^3\), which confirms our earlier assumptions.

By recognizing the pattern in perfect square trinomials, we can identify matches and verify expressions by expanding and comparing terms.

Expansion

The process of expansion involves transforming a compressed or compact form into its extended or full form. When dealing with binomials, it's about expressing something like \((a - b)^2\) in its expanded variant. For example:

• The original \((5x^{\square} - 2)^2\) gives us a compact journal of terms.
  
• Upon expansion, it results in \(25x^6 - 20x^3 + 4\), unraveling individual components of the expression.

Understanding expansion is vital as it allows one to see the detailed breakdown of each term, tracing them back to their simple forms. By performing expansions:

• We get a clearer picture of individual powers and coefficients.
  
• We verify the relationship between combined terms and their sum.

Rewriting equations in expanded form can help verify identities or mechanics behind specific algebraic simplifications.

Exponents

Exponents are mathematical expressions that denote the power to which a number or a variable is raised. In our problem, it involves terms like \(x\) raised to different powers.

• The expression \(5x^{\square}\) on the left side involves an unknown power.
  
• Through solving, we identify that \(5x^3\) aligns with it, clarifying that \(x^{\square} = x^3\).

Exponents play a crucial role:

• They dictate the degree of polynomial terms, influencing their hierarchy.
  
• Recognizing exponent rules helps in expanding, reducing or rearranging equations.

Within polynomials, mastery of exponents is key.

• They transform complex expressions into manageable, comparable components.
  
• This aids in visualization and operational understanding in solving equations.