Problem 59 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(4 x^{2}+y^{2}+3\right)-\left(x^{2}+y^{2}-2\right) $$

Step-by-Step Solution

Answer

3x^2 + 5

Step 1: Distribute the Negative Sign

Rewrite the expression by distributing the negative sign through the second parenthesis: \[(4x^2 + y^2 + 3) - (x^2 + y^2 - 2) \] becomes \[4x^2 + y^2 + 3 - x^2 - y^2 + 2.\]

Step 2: Combine Like Terms

Next, combine like terms from the expression:- Combine the \(x^2\) terms: \(4x^2 - x^2 = 3x^2\)- Combine the \(y^2\) terms: \(y^2 - y^2 = 0\)- Combine the constant terms: \(3 + 2 = 5\)

Step 3: Write the Simplified Expression

After combining like terms, the expression simplifies to:\[3x^2 + 5.\]

Key Concepts

Combining Like Terms

Combining like terms is a fundamental concept in algebra and is crucial for simplifying expressions effectively. "Like terms" refer to terms that contain the same variables raised to the same power. For example, in the expression \(4x^2 - x^2 + y^2 - y^2 + 3 + 2\), the terms \(4x^2\) and \(-x^2\) are considered like terms because they both have the same variable \(x\) raised to the power of 2. Similarly, \(y^2\) and \(-y^2\) are like terms.When combining like terms:

• Add or subtract the coefficients (the numerical part) of the like terms while keeping the variable part unchanged.
• If there are no like terms to combine, the term stays unchanged in the expression.
• In our example, after combining like terms, the \(x^2\) terms become \(3x^2\), the \(y^2\) terms cancel each other to zero, and the constants add up to 5.

Combining like terms helps in reducing an algebraic expression to its simplest form, making it easier to evaluate or solve.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves transforming a given expression into its simplest form by following a series of steps involving operations such as combining like terms and applying the distributive property. In the example \(4x^2 + y^2 + 3 - x^2 - y^2 + 2, \)the goal is to condense this expression to as few terms as possible.Here are the general steps to simplify algebraic expressions:

• First, eliminate any unnecessary parentheses by distributing operations such as subtraction across terms.
• Next, identify and combine all like terms to reduce the number of separate terms in the expression.
• Rearrange and reorder terms if needed, often according to descending powers of variables, for clarity and standard form.

In the example provided, these steps lead us to eliminate \(y^2 - y^2\) because it equals zero, leaving fewer terms to work with. The process of simplifying expressions is key to making complex problems more manageable and strives for the clearest expression possible.

Distributive Property

The distributive property is a vital algebraic concept used when you need to eliminate parentheses in expressions, especially when subtracting or adding terms. This property states that multiplying a sum or difference by a number is the same as multiplying each addend individually and then summing or subtracting the products.Mathematically, this is expressed as:\[a(b + c) = ab + ac.\]In our original exercise, the distributive property is applied to simplify\[(4x^2 + y^2 + 3) - (x^2 + y^2 - 2). \]The negative sign outside the second parenthesis is distributed across each term inside the parentheses, making the expression:

• \(-x^2\)
• \(-y^2\)
• \(+2\)

This results in the transformed expression \(4x^2 + y^2 + 3 - x^2 - y^2 + 2\). Adopting the distributive property allows for proper simplification, effectively preparing the expression for further reduction or solving.