Problem 97 - Intermediate Algebra

Question

Simplify each expression. \(-2\left(3 y^{3}-2 y+7\right)-\left(y^{2}+2 y-4\right)+4\left(y^{3}+2 y-1\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(-2y^3 - y^2 + 10y - 14\).

Step 1: Distribute the Constants

Let's start by distributing the constants over the terms inside the parentheses. For \(-2(3y^3 - 2y + 7)\), distribute \(-2\):\[-2(3y^3) = -6y^3, \ -2(-2y) = +4y, \ -2(7) = -14\]So, the expression becomes:\[-6y^3 + 4y - 14\].Next, distribute the negative sign over \(-(y^2 + 2y - 4)\):\[-(y^2) = -y^2, \ -(2y) = -2y, \ -(-4) = +4\]So, the expression becomes:\[-y^2 - 2y + 4\].Finally, distribute \ 4 \ over \(4(y^3 + 2y - 1)\):\[4(y^3) = 4y^3, \ 4(2y) = 8y, \ 4(-1) = -4\]So, the expression becomes:\[4y^3 + 8y - 4\].

Step 2: Combine Like Terms

Now that we have all the terms distributed, combine like terms from \(-6y^3 + 4y - 14 - y^2 - 2y + 4 + 4y^3 + 8y - 4\):1. Combine the \ y^3 \ terms: \(-6y^3 + 4y^3 = -2y^3\).2. Combine the \ y^2 \ terms: There is only one \ y^2 \ term, \(-y^2\).3. Combine the \ y \ terms: \(4y - 2y + 8y = 10y\).4. Combine the constant terms: \(-14 + 4 - 4 = -14\).Thus the expression simplifies to:\[-2y^3 - y^2 + 10y - 14\].

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps simplify expressions. It allows you to multiply a sum or difference by a number outside of the parentheses. By doing so, you multiply each term inside the parentheses by that number.
For example:

• If you have the expression \(-2(3y^3 - 2y + 7)\), you distribute \(-2\) to each term inside:
  • \(-2 imes 3y^3 = -6y^3\)
  • \(-2 imes -2y = +4y\)
  • \(-2 imes 7 = -14\)

Once the distributive property is applied, the expression changes form, making it easier to combine and simplify using other properties. This method is crucial in managing multi-term expressions effectively.

Like Terms

Like terms are terms in an expression that have the same variable parts. Recognizing like terms is essential because these terms can be combined to simplify equations easily. For example:

• In the expression \(-6y^3 + 4y - 14 - y^2 - 2y + 4 + 4y^3 + 8y - 4\), you can identify the like terms as follows:
  • \(y^3\) terms: \(-6y^3\) and \(4y^3\)
  • \(y^2\) term: \(-y^2\)
  • \(y\) terms: \(4y\), \(-2y\), and \(8y\)
  • Constants: \(-14\), \(+4\), and \(-4\)

By combining these like terms, the expression is greatly simplified. This process reduces error potential and makes complex algebraic work more manageable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a value. Algebra involves transforming and simplifying these expressions to solve equations. In the given exercise, you deal with a multi-term algebraic expression.

Steps to handle algebraic expressions efficiently include:

• Identify all terms and operations involved.
• Apply the distributive property where needed to eliminate parentheses.
• Find and combine like terms to simplify.
• The goal is to reduce the expression to its simplest form.

For example, the expression \(-2(y^3 - y^2) + 4y\) becomes simpler once distributed and like terms are combined. By transforming expressions using these strategies, you develop a better understanding of algebra's logical structure.