Problem 19 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$3\left(n^{2}+1\right)-8\left(n^{2}-1\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-5n^2 + 11\).

Step 1: Distribute the Constants

Start by distributing the constants outside each set of parentheses to both terms inside the parentheses. For the expression \(3(n^2 + 1)\): \[3 \cdot n^2 + 3 \cdot 1 = 3n^2 + 3\]For the expression \(-8(n^2 - 1)\):\[-8 \cdot n^2 + (-8) \cdot (-1) = -8n^2 + 8\]

Step 2: Combine the Distributed Expressions

Combine the two simplified expressions from Step 1 into one expression:\[3n^2 + 3 - 8n^2 + 8\]

Step 3: Combine Like Terms

Combine the like terms in the expression.For the \(n^2\) terms: \[3n^2 - 8n^2 = -5n^2\]For the constant terms:\[3 + 8 = 11\]The expression simplifies to:\[-5n^2 + 11\]

Key Concepts

Polynomials

Polynomials are expressions that consist of variables and coefficients, set in a series of terms added together. Each term has a variable raised to a power, with the coefficient multiplying this term. For example, in the expression \(3n^2\), the number 3 is the coefficient, \(n\) is the variable, and 2 is the exponent. A polynomial can have one or more terms, such as monomials (1 term), binomials (2 terms), and trinomials (3 terms).
Understanding the structure of polynomials is crucial because it helps in performing algebraic operations like addition, subtraction, and multiplication. In the example given, the expression \(3(n^2 + 1)\) is a polynomial expressed in terms that need simplification. Simplifying polynomials often involves applying rules and properties such as the Distributive Property and combining like terms.

• Terms: Units added together in a polynomial.
• Coefficient: Number multiplied by the variable in a term.
• Exponent: Power to which the variable is raised.

Distributive Property

The Distributive Property is a foundational algebraic principle used to simplify expressions, especially when handling parentheses. It states that multiplying a number or term by a group of terms inside parentheses is equal to doing individual multiplications for each term within the parentheses. For example, given \(a(b + c)\), it simplifies to \(ab + ac\).
This property is key when dealing with expressions like \(3(n^2 + 1)\) and \(-8(n^2 - 1)\). It allows the expression to be rewritten without parentheses by distributing the numbers outside, hence multiplying each term inside. For instance:

• \(3(n^2 + 1)\) becomes \(3n^2 + 3\).
• \(-8(n^2 - 1)\) becomes \(-8n^2 + 8\).

By mastering the Distributive Property, you can simplify expressions and set the stage for further combining of like terms.

Combining Like Terms

Combining like terms is a process used to simplify algebraic expressions by grouping and reducing similar terms. Terms are considered like terms if they contain the same variable raised to the same power. In the expression \(3n^2 + 3 - 8n^2 + 8\), the terms \(3n^2\) and \(-8n^2\) are like terms because they share the variable \(n^2\) as their basis.
The main goal is to combine these terms to reduce the expression to its simplest form. Here's how you perform the action:

• Combine \(3n^2\) and \(-8n^2\) to get \(-5n^2\).
• Add the constant terms, 3 and 8, to get 11.

The result of combining like terms in this case is the simplified polynomial \(-5n^2 + 11\). This process helps make complex expressions more manageable and ready for solving or further operations. Such techniques are essential for mastering algebraic manipulation.