Problem 66 - Beginning and Intermediate Algebra

Question

Perform the indicated operations. $$\begin{array}{l} \left(-7 m^{2}-14 m+56\right)+\left(3 m^{2}+7 m-6\right) \\ +\left(9 m^{2}-10\right) \end{array}$$

Step-by-Step Solution

Answer

The simplified expression is: \(5m^2-7m+40\).

Step 1: Identifying the Like Terms

Identify the like terms from all three expressions within the parentheses. \(1^{st}\) Expression: \(-7m^2-14m+56\) \(2^{nd}\) Expression: \(+3m^2+7m-6\) \(3^{rd}\) Expression: \(+9m^2-10\)

Step 2: Combining Like Terms

Combine the like terms for each type of term. For \(m^2\) terms: \[(-7m^2) + (3m^2) + (9m^2)\] For \(m\) terms: \[(-14m) + (7m)\] For constant terms: \[(56) + (-6) + (-10)\]

Step 3: Performing the Operations

Perform the operations identified in Step 2. For \(m^2\) terms: \[(-7+3+9)m^2 = 5m^2\] For \(m\) terms: \[(-14+7)m = -7m\] For constant terms: \(56-6-10 = 40\)

Step 4: Writing out the Simplified Expression

After completing the operations, write out the simplified expression by combining the results. Simplified Expression: \(5m^2-7m+40\)

Key Concepts

Combining Like Terms

When dealing with polynomials, especially in algebraic expressions, it's crucial to understand how to combine like terms. Combining like terms involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. These are called "like terms."

• Identify Like Terms: Before combining, you need to identify which terms are alike. In the expression given, look for terms that have similar variables and powers. For instance, terms with \(m^2\) are grouped together, terms with \(m\) are grouped separately, and constants (terms without variables) stand on their own.
• Add or Subtract: Once identified, carry out the addition or subtraction on the coefficients (the numbers in front of the terms), as shown in the original solution. This helps in simplifying the expression and makes it easier to understand and solve.

Reorganizing terms by identifying and combining like terms helps make complex algebraic expressions more manageable.

Polynomials

Polynomials are algebraic expressions consisting of variables, coefficients, and exponents. They are made up of one or more terms, which can be added, subtracted, multiplied, or divided. Each polynomial is defined by its degree, which is the highest power of the variable present in the expression.

• Terms and Components: Each term in a polynomial includes a coefficient (like \(-7\) or \(3\)), a variable (such as \(m\)), and an exponent. A polynomial can be a single term or several terms together.
• Degree of a Polynomial: The degree is determined by the highest exponent of the variable. In the expression \(5m^2 - 7m +40\), the degree is 2 because the highest power of \(m\) is 2.

Understanding polynomials is fundamental in algebra as they form the basis for expressing equations and functions in mathematics.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and several operations (like addition, subtraction, etc.). These expressions allow us to create equations that model real-world situations and complex mathematical relationships.

• Variables and Constants: In an algebraic expression, variables (like \(m\)) represent unknowns that we often need to solve for. Constants are the numbers on their own, without any variables attached.
• Operations: By using operations between terms, you can transform and solve algebraic expressions. In the process of simplifying expressions or solving equations, you'll typically perform polynomial operations to combine like terms, factor, or expand.

Algebraic expressions are versatile tools in mathematics, enabling us to describe patterns and relationships concisely and clearly.