Problem 106 - Intermediate Algebra

Question

Perform each operation. $$ \left(3 c^{2}+5 c\right)+\left(7-c^{2}-5 c\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(2c^2 + 7\).

Step 1: Rewrite with parentheses for clarity

We begin by rewriting the expression with parentheses to clearly see each group that will be combined:\[ (3c^2 + 5c) + (7 - c^2 - 5c) \] This helps us organize terms before addressing addition.

Step 2: Remove parentheses using distribution

The expression inside the parentheses doesn't need multiplication distribution since all terms are being added or subtracted directly. So remove the parentheses:\[ 3c^2 + 5c + 7 - c^2 - 5c \]

Step 3: Combine like terms for each variable

Identify and combine like terms separately for each type of term (constant terms, terms with \(c\), terms with \(c^2\)):\[ (3c^2 - c^2) + (5c - 5c) + 7 \]This combines to:\[ 2c^2 + 0c + 7 \]

Step 4: Simplify expression

Simplify the expression by removing zero coefficient terms, which in this case is \(0c\):\[ 2c^2 + 7 \]This simplifies the final expression.

Key Concepts

Combining Like Terms

One of the core aspects of working with polynomials is understanding how to combine like terms. Like terms are terms that have the exact same variable raised to the same power. For example, in the expression \(3c^2 + 5c + 7 - c^2 - 5c\), the terms \(3c^2\) and \(-c^2\) are like terms because they both involve \(c^2\). Similarly, \(5c\) and \(-5c\) are like terms since they both involve \(c\).

Combining these terms means we perform addition or subtraction to reduce the expression. From our example above, combining \(3c^2\) and \(-c^2\) results in \(2c^2\), and \(5c\) combined with \(-5c\) results in \(0c\), which is essentially zero.

To effectively combine like terms, remember to:

• Identify terms within the expression that have the same variables and powers.
• Add or subtract the coefficients of these terms as appropriate.
• Simplify the expression by removing terms with a coefficient of zero.

Simplifying Polynomials

Simplifying polynomials involves reducing them to their most condensed form by combining like terms and eliminating unnecessary terms. After combining like terms in an expression such as \(3c^2 - c^2 + 5c - 5c + 7\), you are left with \(2c^2 + 0c + 7\).

Since a term with a zero coefficient doesn't affect the value, \(0c\) can be removed, leaving \(2c^2 + 7\). Simplified expressions are easier to work with and interpret in subsequent algebraic operations.

Some tips for simplifying polynomials include:

• Ensure all like terms are combined first.
• Look for zero coefficient terms and omit them once combined.
• Constant numbers (without variables) typically cannot be simplified further unless combined with another constant.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operational symbols. In the expression \((3c^2 + 5c) + (7 - c^2 - 5c)\), we have a mix of constants (like 7), and variables (like \(c\)) to various powers.

Working with algebraic expressions involves the ability to manipulate and alter them while still maintaining their equivalence. This manipulation includes adding, subtracting, and simplifying as necessary, such as we do with polynomials.

Understanding and breaking down algebraic expressions is crucial because it forms the basis of solving equations and understanding higher-level math concepts. Here's how you can manage them effectively:

• Identify and group like terms to simplify calculations.
• Pay attention to the order of operations, especially when dealing with parentheses.
• Ensure clarity and precision to minimize errors.