Problem 27 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(15-34\) by removing parentheses and combining similar terms. $$-2\left(n^{2}-4\right)-4\left(2 n^{2}+1\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-10n^{2} + 4\).

Step 1: Expand each term with parentheses

Start by expanding the terms inside the parentheses by distributing the multiplication over addition or subtraction. For \(-2(n^{2}-4)\), distribute \(-2\) across \(n^{2}\) and \(-4\):\[-2(n^{2}) - 2(-4) = -2n^{2} + 8\].For \(-4(2n^{2}+1)\), distribute \(-4\) across \(2n^{2}\) and \(1\):\[-4(2n^{2}) - 4(1) = -8n^{2} - 4\].This gives us the expanded expression: \(-2n^{2} + 8 - 8n^{2} - 4\).

Step 2: Combine like terms

Identify like terms in the expanded expression to combine them. In \(-2n^{2} + 8 - 8n^{2} - 4\), the like terms are the \(n^{2}\) terms: \(-2n^{2}\) and \(-8n^{2}\). Combine these to get:\[-2n^{2} - 8n^{2} = -10n^{2}\].Next, combine the constant terms \(+8\) and \(-4\):\[8 - 4 = 4\].Thus, the simplified expression is \(-10n^{2} + 4\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that involves distributing a single factor across terms inside parentheses. It essentially uses multiplication over addition or subtraction.

Imagine you have an expression like \(-2(n^2-4)\). Instead of solving inside the parentheses first, you multiply \(-2\) by each term inside. This means:

• Multiply \(-2\) with \(n^2\) to get \(-2n^2\).
• Multiply \(-2\) with \(-4\) to get \(+8\) (note the change of sign because multiplying two negatives gives a positive).

Similarly, for \(-4(2n^2+1)\), you distribute \(-4\):

• Multiply \(-4\) with \(2n^2\) to get \(-8n^2\).
• Multiply \(-4\) with \(1\) to get \(-4\).

The result is a new expression where the parentheses have been "removed" through distribution, leading us to tackle further simplifications.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power.

In our expanded expression \-2n^2 + 8 - 8n^2 - 4\, we need to spot these.

The term \(-2n^2\) is similar to \(-8n^2\) because they both contain \(n^2\). Combine them by adding their coefficients together: \(-2 - 8 = -10\). This gives us \-10n^2\.

Next, we look at the constant terms. These are numbers without variables. In this case: \(8\) and \(-4\). When added together, they simplify to \(4\).

This process of combining like terms reduces the expression to fewer terms, making it easier to interpret and solve.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients combined using addition, subtraction, and multiplication. They are versatile and show up frequently in various forms across algebra.

Polynomials are classified based on their degree and the number of terms:

• A monomial has one term like \(-10n^2\).
• A binomial has two terms, for instance, \(-10n^2 + 4\).
• A trinomial has three terms, such as \(3x^2 - 2x + 1\).

A polynomial's degree is determined by the highest power of the variable. In \(-10n^2 + 4\), the degree is \(2\) because the highest power of \(n\) is \(2\).

Understanding polynomial expressions helps in identifying how to apply operations such as the distributive property and combination of like terms to simplify or solve them. It's crucial to grasp their structure for more advanced algebraic maneuvers.