Problem 12 - College Algebra with Modeling and Visualization

Question

Combine like terms whenever possible. $$5 a b-b^{2}+7 a b+6 b^{2}$$

Step-by-Step Solution

Answer

The expression simplifies to \(12ab + 5b^2\).

Step 1: Identify Like Terms

First, we need to identify which terms in the expression are like terms. Like terms are those that have the same variable parts. In the given expression, the like terms are: 1. \(5ab\) and \(7ab\)2. \(-b^2\) and \(6b^2\).

Step 2: Combine Like Terms for \(ab\)

Add the coefficients of the \(ab\) terms together. The terms are \(5ab\) and \(7ab\), so:\[5ab + 7ab = 12ab\].

Step 3: Combine Like Terms for \(b^2\)

Add the coefficients of the \(b^2\) terms together. The terms are \(-b^2\) and \(6b^2\), so:\[-b^2 + 6b^2 = (6-1)b^2 = 5b^2\].

Step 4: Write the Combined Expression

Combine the results from Step 2 and Step 3 into the final expression. The expression after combining like terms is:\[12ab + 5b^2\].

Key Concepts

Like Terms

In algebra, **like terms** are terms that contain the same variables raised to the same powers. This means they match exactly in their variable parts, not just in the type of variables they have or the order of multiplication. Identifying like terms is crucial in simplifying algebraic expressions because it tells us which terms we can combine.

In our example, the expression is:

• \( 5ab \)
• \( -b^2 \)
• \( 7ab \)
• \( 6b^2 \)

If we look closely, the like terms are:

• \( 5ab \) and \( 7ab \)
• \( -b^2 \) and \( 6b^2 \).

Recognizing these allows us to perform the next step of combining them. Always check the whole part of the term before concluding that they are like terms.

Combining Terms

Once we've identified like terms, **combining them** is the next step, which involves adding or subtracting their coefficients while keeping the variable part unchanged. This process streamlines the expression, making it easier to handle in future calculations.

For instance, in our expression:- The terms \( 5ab \) and \( 7ab \) can be combined because they're like terms. Add their coefficients: \[ 5 + 7 = 12 \] So, \( 5ab + 7ab = 12ab \).- Next, for the terms \( -b^2 \) and \( 6b^2 \), combine like this: \[ -1 + 6 = 5 \] Therefore, \(-b^2 + 6b^2 = 5b^2\).

Both operations preserve the variables' integrity, only affecting their coefficients, leading us smoothly to the final expression.

Polynomial Simplification

**Polynomial simplification** is the process of reducing a polynomial to its simplest form by combining like terms. This is essential for making the expression easier to evaluate, compare, or solve. A simplified polynomial expression is often more intuitive and manageable.

In the given example, we started with a polynomial expression:\[ 5ab - b^2 + 7ab + 6b^2 \]By identifying and combining like terms, we've reduced it to:\[ 12ab + 5b^2 \]This simpler expression is equivalent to the original. Simplification involves no change to the values involved, only how they are presented. It is fundamental in calculations involving algebraic expressions, either for solving equations or evaluating them in specific scenarios. Thus, mastering simplification skills aids in both understanding and solving algebraic problems more efficiently.