Problem 15 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ \left(3 x^{2}+x+1\right)+\left(2 x^{2}-3 x-5\right) $$

Step-by-Step Solution

Answer

\(5x^2 - 2x - 4\)

Step 1: Identify the Expression Components

The expression we need to simplify is \( (3x^2 + x + 1) + (2x^2 - 3x - 5) \). Note that it consists of two polynomials inside parentheses.

Step 2: Distribute the Addition

We add the two polynomials together by removing the parentheses: \( 3x^2 + x + 1 + 2x^2 - 3x - 5 \).

Step 3: Combine Like Terms

Combine the terms with the same powers of \( x \). For \( x^2 \) terms: \( 3x^2 + 2x^2 = 5x^2 \). For \( x \) terms: \( x - 3x = -2x \). For constant terms: \( 1 - 5 = -4 \).

Step 4: Write the Simplified Expression

The simplified expression is obtained after combining like terms: \( 5x^2 - 2x - 4 \).

Key Concepts

Like Terms

In algebra, the concept of "like terms" is essential for simply managing equations, especially when it comes to polynomial addition. Like terms are terms that have the same variable raised to the same power. For instance, in the expression \( (3x^2 + x + 1) + (2x^2 - 3x - 5) \), like terms are those that both include \( x^2 \), both include \( x \), and both are constants. Recognizing like terms allows us to combine terms to simplify expressions. For example:

• \( 3x^2 \) and \( 2x^2 \) are like terms because they both contain \( x^2 \).
• \( x \) and \( -3x \) are like terms because they both contain \( x \).
• 1 and -5 are constants, hence they are like terms too.

By organizing and combining these like terms, the process becomes much more straightforward, leading to simpler expressions and easier-to-understand results.

Simplifying Expressions

Simplifying expressions involves the process of reducing them into their simplest form. This is done by combining like terms and performing basic arithmetic operations. When faced with an expression such as \( 3x^2 + x + 1 + 2x^2 - 3x - 5 \), simplifying is necessary for clarity and efficiency.Here are the steps to simplify this expression:

• **Step 1:** Identify all like terms in the expression, grouping them together.
• **Step 2:** Add or subtract these like terms. For instance, combine \( 3x^2 \) and \( 2x^2 \) to get \( 5x^2 \).
• **Step 3:** Perform similar operations for the linear terms \( x \) and \( -3x \), and for the constants 1 and -5.

After combining all the like terms, you'll achieve a streamlined expression: \( 5x^2 - 2x - 4 \). This simplified form makes it easier to interpret and use in further calculations.

Combined Polynomials

Understanding combined polynomials is key when performing polynomial addition. When you combine polynomials, you essentially merge all of their terms into a unified expression. In our exercise, the given polynomials are \( 3x^2 + x + 1 \) and \( 2x^2 - 3x - 5 \).Here’s how you handle combined polynomials:

• First, take each term of one polynomial and add it to the corresponding like term from the other polynomial.
• For example, combine the \( x^2 \) terms from each polynomial: \( 3x^2 \) from the first polynomial and \( 2x^2 \) from the second, resulting in \( 5x^2 \).
• Continue this pattern with the linear terms (\( x \) terms) and the constant terms.

Through combining, the resulting polynomial \( 5x^2 - 2x - 4 \) is a more consolidated form that expresses the full contribution of both original polynomials. This process is crucial in algebra, allowing for the integration of multiple expressions into a single, unified polynomial.