Problem 59 - Algebra for College Students

Question

Simplify by removing the inner parentheses first and working outward. $$2 x^{2}-\left[-3 x^{2}-\left(x^{2}-4\right)\right]$$

Step-by-Step Solution

Answer

The expression simplifies to \(6x^2 - 4\).

Step 1: Remove the Innermost Parentheses

Begin by simplifying the expression inside the innermost parentheses: \( x^2 - 4 \), which remains the same as there are no operations to perform within it. The expression now is:\[2x^2 - \left[-3x^2 - (x^2 - 4)\right]\]

Step 2: Distribute and Simplify the Middle Parentheses

Now, distribute the negative sign across the expression within the middle parentheses:\[-3x^2 - x^2 + 4\]Combine like terms inside the middle parentheses:\[-4x^2 + 4\]Replace the middle expression in the overall expression, resulting in:\[2x^2 - (-4x^2 + 4)\]

Step 3: Remove the Outer Parentheses

Eliminate the outer parentheses by distributing the negative sign again:\[2x^2 + 4x^2 - 4\]

Step 4: Combine Like Terms

Combine the like terms in the expression:\[(2x^2 + 4x^2) - 4 = 6x^2 - 4\]

Step 5: Final Simplified Expression

The final simplified expression is: \[6x^2 - 4\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to remove parentheses by distributing a multiplication over addition or subtraction inside the parentheses. In our example, we have multiple layers of parentheses to work through.

• In the first instance, we used the property to distribute a negative sign across the inner contents: \[-3x^2 - x^2 + 4\]
• Here, it was crucial to remember that distributing a negative is akin to multiplying by \(-1\), which changes the signs of each term inside the parentheses.
• Later on, we applied the distributive property again, this time on the expression \(2x^2 - (-4x^2 + 4)\), which turned into: \[2x^2 + 4x^2 - 4\]

Understanding how to use the distributive property ensures that expressions are simplified accurately and efficiently.

Combining Like Terms

Combining like terms is another key concept in polynomial simplification. It involves merging terms in an algebraic expression that share the same variable raised to the same power.

• In the expression \(-3x^2 - x^2 + 4\), both \(-3x^2\) and \(-x^2\) are like terms because they both contain \(x^2\).
• We can combine these to get \(-4x^2\).
• Similarly, in \(2x^2 + 4x^2 - 4\), the terms \(2x^2\) and \(4x^2\) can be combined to form \(6x^2\).

Always look out for terms with identical variables and exponents. Group them to streamline the expression into a simpler form.

Nested Parentheses

Nested parentheses require careful attention as they involve handling multiple layers of expressions wrapped in brackets. These can sometimes appear intimidating, but their systematic simplification is essential.

• Start from the innermost parentheses and work outward, one layer at a time.This provides a clear path through the expression.
• For example, begin by resolving \(x^2 - 4\), then shift focus to distributing within \([-3x^2 - (x^2 - 4)]\), and finally managing the entire expression \(2x^2 - [-4x^2 + 4]\).

Progressing methodically through each layer ensures that no detail is overlooked and the simplification process remains structured.

Polynomial Expressions

Polynomials are algebraic expressions made up of terms involving variables raised to whole number exponents and their coefficients. Simplifying polynomials is an important skill, as it often makes the expressions easier to work with.

• The final simplified expression in the given problem is \[6x^2 - 4\].
• This contains two terms: \(6x^2\), which is a polynomial term, and \(-4\), which is a constant term.
• Each term in a polynomial is typically separated by a plus or minus sign.

Working with polynomials involves recognizing the structure of these terms and applying simplification techniques like removing parentheses and combining like terms.