Problem 49 - Algebra A Combined Function

Question

Add \(\left(4 x^{2}-6 x+1\right)\) and \(\left(3 x^{2}+2 x+1\right)\)

Step-by-Step Solution

Answer

The sum is \(7x^2 - 4x + 2\).

Step 1: Identify Like Terms

Write down the expressions you need to add: \(4x^2 - 6x + 1\) and \(3x^2 + 2x + 1\). Identify which terms are like terms (terms with the same variable and exponent). The like terms here are \(4x^2\) with \(3x^2\), \(-6x\) with \(2x\), and the constants (\(1\) and \(1\)).

Step 2: Add the Like Terms Separately

To combine the expressions, add the coefficients of the like terms together. - For \(x^2\) terms: \(4 + 3 = 7\), so the term becomes \(7x^2\). - For \(x\) terms: \(-6 + 2 = -4\), so the term becomes \(-4x\). - For constants: \(1 + 1 = 2\).

Step 3: Write the Final Expression

Combine the results from Step 2 to form the final expression. The equation becomes \(7x^2 - 4x + 2\).

Key Concepts

Like Terms

When dealing with polynomials, the concept of 'like terms' is fundamental. Like terms are terms that share identical variables raised to the same power. In our expression
addition example \( (4x^2 - 6x + 1) + (3x^2 + 2x + 1) \), we identify like terms by examining each polynomial part.

• The term \( 4x^2 \) from the first polynomial and \( 3x^2 \) from the second are like terms because both consist of \( x^2 \).
• The terms \( -6x \) and \( 2x \) are similar, as they both include the \( x \) variable raised to the first power.
• Lastly, the constants (1 and 1) are like terms as they do not contain any variables.

Understanding the concept of like terms allows you to combine similar parts easily, making operations on polynomials more manageable.

Simplifying Expressions

Simplifying expressions is all about combining like terms to reduce a polynomial to its simplest form. It involves performing operations on terms and ensuring that the expression is concise without altering its value.

In the \( 4x^2 - 6x + 1 + 3x^2 + 2x + 1 \) example, we simplify by first grouping like terms:

• Combine \( 4x^2 \) and \( 3x^2 \) to get \( 7x^2 \).
• Combine \( -6x \) and \( 2x \) to get \( -4x \).
• Add the constants \( 1 \) and \( 1 \) to yield \( 2 \).

After combining these like terms, the expression simplifies to\( 7x^2 - 4x + 2 \).
This process not only organizes the polynomial neatly but also makes further algebraic manipulations much easier.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and arithmetic operations. In an algebraic context, understanding how to manipulate these expressions is key to mastering algebra.

An expression like \( 4x^2 - 6x + 1 \) is formed using:

• Terms, which here are \( 4x^2 \), \( -6x \), and \( 1 \).
• Operations, such as addition and subtraction that connect these terms.
• Variables, like \( x \), represent unknown quantities and are the focus of algebraic manipulation.

Such expressions can be simplified by recognizing like terms and following a routine process of addition or subtraction. These skills lay the groundwork for solving more complex problems, as mastering algebraic expressions is crucial for higher-level mathematics.