Problem 21 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ -6\left(x^{2}-5\right)-\left(x^{2}-2\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-7x^2 + 32\).

Step 1: Distribute the Negative Sign

The expression is given as \[-6(x^2-5)-(x^2-2)\].First, distribute the negative sign in front of the second parenthesis. This means multiply -1 with each term inside the parentheses: \[-6(x^2-5) - x^2 + 2\] which simplifies to \[-6(x^2-5) - x^2 + 2\].

Step 2: Distribute and Simplify

Now, distribute the -6 through the first parenthesis:\[-6 \cdot x^2 - 6 \cdot (-5) - x^2 + 2\].This simplifies to:\[-6x^2 + 30 - x^2 + 2\].

Step 3: Combine Like Terms

Combine the terms with the same variable:For the \(x^2\) terms: \[-6x^2 - x^2 = -7x^2\].All constants: \[30 + 2 = 32\]. Thus, the simplified expression becomes \[-7x^2 + 32\].

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operational symbols like addition, subtraction, multiplication, and division. Think of them as sentences in algebra. They might involve simple variables like \(x\) and \(y\) or constants and coefficients. An algebraic expression can stand for various quantities, and we manipulate these expressions to determine unknown values or to simplify them.

In general, expressions can be quite complex, involving exponents, multiple terms, or even fractions. A key part of working with these expressions is understanding how to rearrange and simplify them without changing their value. By organizing terms of similar kinds and reducing overall complexity, we make expressions easier to work with, especially when solving algebraic equations or evaluating mathematical scenarios. This simplification often makes further operations more manageable and less error-prone. When you see a big, overwhelming expression, remember that your goal is to distill it down to its simplest form.

Combining Like Terms

Combining like terms is a basic principle in algebra that helps in simplifying expressions and making them more manageable. In algebraic expressions, 'like terms' refer to terms that have identical variable parts. For example, in the expression \(4x + 3xy - 5x + x^2\), the terms \(4x\) and \(-5x\) are like terms because they both contain the variable \(x\) raised to the same power.

To combine these, we simply add or subtract their coefficients. Thus, combining \(4x\) and \(-5x\) results in \(-x\). This method also extends to constant terms, like combining numbers without variables such as \(30 + 2 = 32\).

• Identify like terms—terms with the same variable and exponent.
• Add or subtract the coefficients of these terms while keeping the variable part unchanged.
• Simplify the expression by reducing the number of terms.

Remember, simplifying through combining like terms reduces clutter and helps reveal the expression's true form. It’s an essential skill that supports everything from solving equations to graphing functions.

Distributive Property

The distributive property is a fundamental algebraic concept used to multiply a single term across a sum or difference inside parentheses. It states that \(a(b + c) = ab + ac\). This property allows us to "distribute" multiplication over addition or subtraction inside parentheses, turning complex expressions into simpler, expanded forms.

When you encounter an expression like \(-6(x^2 - 5)\), you use the distributive property to multiply \(-6\) with each term inside the parentheses:

• Multiply \(-6\) by \(x^2\) to get \(-6x^2\).
• Multiply \(-6\) by \(-5\) to get \(30\).

This gives a simplified form: \(-6x^2 + 30\).

Using the distributive property effectively helps break down expressions into their component parts, making subsequent operations like combining like terms more straightforward. It's a versatile tool that not only aids in expression simplification but is also useful in solving equations and inequalities, manipulating formulas, and expanding polynomial expressions.