Problem 40 - Elementary Algebra

Question

State the restrictions and then simplify. $$ 16 x 2-1(4 x+1) 2 $$

Step-by-Step Solution

Answer

The simplified expression is \(-8x - 1\), and there are no restrictions on \(x\).

Step 1: Identify the Expression

The given expression is \(16x^2 -1(4x+1)^2\), which needs to be simplified and any restrictions on the variable \(x\) identified.

Step 2: Expand the Second Term

First, expand the second term \(-1(4x+1)^2\). Calculate \((4x+1)^2\) as it equals \((4x+1)(4x+1)\), resulting in \(16x^2 + 8x + 1\). Multiply this by -1 to get \(-16x^2 - 8x - 1\).

Step 3: Simplify the Expression

Substitute the expanded form into the original expression: \(16x^2 - 16x^2 - 8x - 1\). Simplify by combining like terms. \(16x^2 - 16x^2\) equals 0, so the expression simplifies to \(- 8x - 1\).

Step 4: State Restrictions

There are no denominators in this expression, which typically introduce restrictions. Therefore, there are no restrictions on \(x\), meaning \(x\) can take any real number.

Key Concepts

Expression Simplification

Simplifying algebraic expressions means reducing them to their simplest form. It requires you to combine like terms and apply mathematical operations carefully.
To simplify expressions like the one in the exercise, follow these steps:

• *Identify like terms:* These are terms with the same variables and powers.
• *Perform operations:* Apply addition, subtraction, multiplication, or division as needed.
• *Simplify:* Combine like terms or numerical coefficients to make the expression as simple as possible.

For example, in the expression given \(16x^2 - 1(4x+1)^2\), the aim is to break down and reorganize terms. In step 3 of the exercise, terms like \(16x^2 - 16x^2\) are combined to result in zero, simplifying the expression further to \(-8x - 1\).
This process helps in solving equations more easily in later calculations.

Variable Restrictions

Variable restrictions occur when certain values of a variable lead to undefined expressions. This often happens with fractions or square roots, such as having zero in the denominator or negative numbers in a square root.
However, in our exercise \(16x^2 - 1(4x+1)^2\), it was noted that there are no fractions or square roots that could cause restrictions. This means the variable \(x\) isn't restricted and can be any real number.
In different contexts, if the expression had been in the form of \(\frac{1}{x-1}\), for example, \(x\) couldn't be 1, as that would make the denominator zero. Always check denominators and square roots in your expressions.

Polynomial Expansion

The polynomial expansion involves breaking down an expression involving brackets, like \((4x + 1)^2\).
This requires multiplying every term inside the brackets by each other to "expand" the expression.

• Start by identifying the terms, here it is \((4x + 1)\).
• Multiply each term: \((4x + 1)(4x + 1)\) yields \(16x^2 + 8x + 1\).

This step increases the number of terms temporarily, often requiring further simplification, as seen in our exercise.
Multiplying by -1 reorganized the expanded terms into \(-16x^2 - 8x - 1\). Polynomial expansion is crucial for later steps in algebraic simplification.