Problem 62 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(a^{2}-a b+4 b^{2}\right)+\left(6 a^{2}+8 a b-b^{2}\right) $$

Step-by-Step Solution

Answer

The result is \( 7a^{2} + 7ab + 3b^{2} \).

Step 1: Write Down the Expression

First, we clearly identify the given expression: \( \left( a^{2} - ab + 4b^{2} \right) + \left( 6a^{2} + 8ab - b^{2} \right) \). It consists of two polynomials that we need to add.

Step 2: Remove the Parentheses

Since we are dealing with an addition of two expressions, we can remove the parentheses: \( a^{2} - ab + 4b^{2} + 6a^{2} + 8ab - b^{2} \).

Step 3: Group Like Terms

Next, group the like terms together, which makes it easier to simplify: \( (a^{2} + 6a^{2}) + (-ab + 8ab) + (4b^{2} - b^{2}) \).

Step 4: Simplify Each Group of Like Terms

Add or subtract the coefficients of the like terms: 1. \( a^{2} + 6a^{2} = 7a^{2} \), 2. \(-ab + 8ab = 7ab\), 3. \( 4b^{2} - b^{2} = 3b^{2} \).

Step 5: Write the Final Answer

Combine the simplified terms to write the final polynomial: \( 7a^{2} + 7ab + 3b^{2} \).

Key Concepts

Like Terms

In algebra, when working with polynomials, one of the most fundamental principles is identifying and working with "like terms." Like terms are terms that have exactly the same variables raised to the same power. This means the only difference between these terms is their coefficients. Understanding this concept allows you to combine terms efficiently. For example, in the exercise provided: both \( a^2 \) from \( a^2 - ab + 4b^2 \) and \( 6a^2 \) from \( 6a^2 + 8ab - b^2 \) are like terms because they both contain the variable \( a \) squared.

• To identify like terms, look for the same variable powers.
• These terms can be combined by adding or subtracting their coefficients.
• Recognizing like terms simplifies polynomial expressions and helps in performing operations like addition or subtraction correctly.

By grouping like terms together, it allows you to simplify the expression by handling each group of like terms separately.

Coefficients

A crucial component in understanding and manipulating algebraic expressions is the concept of coefficients. A coefficient is the numerical factor in a term of an algebraic expression. In other words, it is the number that is multiplied by the variable or variables of the term. For instance, in the term \( 6a^2 \), the coefficient is 6. Understanding coefficients is essential because they determine the size or amount of the term they are adjacent to.

• Coefficients can be positive or negative integers, fractions, or decimals.
• When adding or subtracting like terms, you simply operate on the coefficients while keeping the variable part unchanged.
• In our exercise, \( a^2 + 6a^2 = 7a^2 \) because you add the coefficients 1 (implied) and 6, but the variable part \( a^2 \) remains the same.

Manipulating coefficients correctly is key to accurately combining like terms in polynomials and ensures the final expression is complete and precise.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations like addition, subtraction, multiplication, and division. They are the basic structures used to describe real-world problems and mathematical concepts using algebra. In the provided exercise, we worked with two algebraic expressions: \( (a^2 - ab + 4b^2) \) and \( (6a^2 + 8ab - b^2) \). Each expression is a combination of terms that are either added or subtracted. Understanding algebraic expressions involves knowing how to interpret these terms and how to perform operations on them.

• Each term in an algebraic expression can have a variable and a coefficient.
• Operations on algebraic expressions often involve combining like terms.
• Simplifying expressions is a way to make them easier to work with, and typically involves reducing the number of terms.

By breaking down algebraic expressions into parts, you can perform operations like addition and subtraction systematically to solve problems and obtain concise results, as demonstrated in the exercise.