Understanding the Expression Simplification

The problem asks us to simplify the algebraic expression \( (4x + 1)^2 - (4x + 3)(4x - 1) \). To do this, we need to expand the terms and then combine like terms. This involves using algebraic identities and careful calculation.

Step-by-Step Simplification Process

Let's break down the simplification into steps:

1. Expand the first term: \( (4x + 1)^2 \)

   We use the algebraic identity \( (a+b)^2 = a^2 + 2ab + b^2 \). Here, \(a = 4x\) and \(b = 1\).

   \( (4x + 1)^2 = (4x)^2 + 2(4x)(1) + 1^2 \)

   \( = 16x^2 + 8x + 1 \)

2. Expand the second term: \( (4x + 3)(4x - 1) \)

   We can use the distributive property (often called FOIL) or recognize a pattern similar to \( (a+b)(a-c) \) or simply expand it step by step:

   • First terms: \( (4x)(4x) = 16x^2 \)
   • Outer terms: \( (4x)(-1) = -4x \)
   • Inner terms: \( (3)(4x) = 12x \)
   • Last terms: \( (3)(-1) = -3 \)

   Adding these together:

   \( (4x + 3)(4x - 1) = 16x^2 - 4x + 12x - 3 \)

   \( = 16x^2 + 8x - 3 \)

3. Subtract the second expanded term from the first

   Now we substitute the expanded forms back into the original expression:

   \( (16x^2 + 8x + 1) - (16x^2 + 8x - 3) \)

   Remember to distribute the negative sign to every term inside the second parenthesis:

   \( = 16x^2 + 8x + 1 - 16x^2 - 8x + 3 \)

4. Combine like terms

   Group the terms with \(x^2\), terms with \(x\), and constant terms:

   \( = (16x^2 - 16x^2) + (8x - 8x) + (1 + 3) \)

   \( = 0x^2 + 0x + 4 \)

   \( = 4 \)

The simplified expression is \( 4 \).

Verification of Simplification

Let's quickly check our steps and calculations. The expansion of \( (4x+1)^2 \) is correct. The expansion of \( (4x+3)(4x-1) \) using FOIL is also correct. The subtraction and combining of like terms resulted in the cancellation of the \(x^2\) and \(x\) terms, leaving only the constant term.

The result of the simplification is indeed 4.

Revision Table: Key Algebraic Identities

Additional Information: Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. It involves rewriting an expression in a simpler or more manageable form. This often requires applying properties of numbers, exponent rules, and algebraic identities.

• Combining Like Terms: Terms that have the same variables raised to the same powers can be added or subtracted. For example, \(3x^2\) and \(5x^2\) are like terms, but \(3x^2\) and \(5x\) are not.
• Distributive Property: \(a(b+c) = ab + ac\). This property is used extensively when expanding products.
• Order of Operations: When simplifying expressions, follow the order of operations (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
• Factoring: Sometimes simplification involves factoring expressions into products of simpler ones, which is the reverse of expansion.

In this specific problem, we used expansion and combining like terms to arrive at the simplified form. The expression simplified to a constant value, meaning its value does not depend on the variable \(x\).