Problem 21 - Algebra 2

Question

Simplify. $$ \left(3 x^{2}-x+2\right)+\left(x^{2}+4 x-9\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(4x^2 + 3x - 7\).

Step 1: Distribute the Addition

Remove the parentheses, since the addition of the two expressions does not change the signs of any of the terms:\[(3x^2 - x + 2) + (x^2 + 4x - 9) = 3x^2 - x + 2 + x^2 + 4x - 9\]

Step 2: Group Like Terms

Identify and group like terms, which are the terms that have the same variable raised to the same power:\[3x^2 + x^2 - x + 4x + 2 - 9\]

Step 3: Combine Like Terms

Add or subtract the coefficients of the grouped like terms:- Combine the \(x^2\) terms: \(3x^2 + x^2 = 4x^2\)- Combine the \(x\) terms: \(-x + 4x = 3x\)- Combine the constant terms: \(2 - 9 = -7\)Thus, the expression simplifies to:\[4x^2 + 3x - 7\]

Key Concepts

Like Terms

When working with polynomials, identifying like terms is crucial. Like terms are terms that contain the same variable raised to the same power. In simpler words, both the letter and its exponent must match for two terms to be like terms. In the expression \(3x^2 - x + 2 + x^2 + 4x - 9\), like terms are grouped together based on their shared characteristics.

Here's a helpful way to spot them:

• Look for terms that have the same base: like \(x^2\) or just \(x\).
• Check if the exponents match for those variables.
• Constant terms, which are standalone numbers without variables, are also like terms amongst themselves.

Recognizing and grouping like terms correctly allows you to combine them effectively in later steps of simplifying expressions.

Distribute

In polynomial addition, distributing often refers to the process of removing parentheses, especially when dealing with addition or subtraction of terms. This step is about ensuring that each term in the parentheses is treated separately.

When you distribute addition, you're essentially reorganizing the expression so that the parentheses no longer change the terms' signs. For example, in \((3x^2 - x + 2) + (x^2 + 4x - 9)\), once the parentheses are removed, each term stands independently:

\[3x^2 - x + 2 + x^2 + 4x - 9\]

It's crucial to note that while distributing addition doesn't affect the signs, distributing subtraction or multiplication would require more consideration of how the signs change.

Combine Coefficients

Once you have identified like terms, the next step is to combine them. This involves adding or subtracting their coefficients. The coefficient of a term is the numeric part that multiplies the variable. By focusing only on these numeric parts, you simplify the expression.

For the exercise, let's break it down:

• The terms \(3x^2\) and \(x^2\) are combined by adding their coefficients: \(3 + 1 = 4\), leading to \(4x^2\).
• The \(x\) terms are \(-x\) and \(4x\). Their coefficients, \(-1\) and \(4\), combine to make \(3x\).
• Finally, for constant terms \(2\) and \(-9\), you perform \(2 - 9 = -7\).

By concentrating on these numeric parts, the expression becomes neater and more straightforward to interpret.

Simplify Expression

Simplifying an expression means making it as compact and readable as possible. It's the culmination of combining like terms and distributing any operations across the terms.

In this exercise, the simplified expression was reached after combining the like terms. This process produced a streamlined expression without any redundant complexity.

The transformations gave us:

• \(3x^2 + x^2 = 4x^2\)
• \(-x + 4x = 3x\)
• \(2 - 9 = -7\)

This resulted in the final simplified expression: \(4x^2 + 3x - 7\).

Remember, simplifying is about finding the cleanest expression that still accurately represents the original problem. It's key to keeping your calculations error-free and your solutions clear.