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Chapter 4: Problem 6

Use the distributive property to remove parentheses. $$-3\left(y^{2}+2\right)$$

Short Answer

Expert verified
The expression \(-3(y^{2}+2)\), when simplified using the distributive property, results in \(-3y^{2} -6\).

Step by step solution

01

Prepare to Distribute

Start with the given expression: \(-3(y^{2}+2)\). The distributive property allows the -3 to be distributed to each term within the parentheses.
02

Apply the Distributive Property

Next, multiply -3 by \(y^{2}\) and -3 by 2. The resulting terms will be \(-3y^{2}\) and -6, respectively.
03

Write the Final Result

Combine the terms from the previous step to get the final expression: \(-3y^{2} -6\). This expression is what we get when we remove the parentheses using the distributive property.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
Algebraic expressions are a fundamental part of mathematics. They consist of numbers, variables, and arithmetic operations like addition and multiplication.
For instance, in the expression \(-3(y^{2}+2)\), each part has a specific role:
  • \(-3\) is a coefficient that multiplies the terms inside the parentheses.
  • \(y^{2}\) is a variable term, representing a variable raised to the power of 2.
  • 2 is a constant term within the parentheses.
Together, these elements form an expression that can be modified or simplified depending on the operation used, like the distributive property.
Polynomials
When we discuss polynomials, we mean algebraic expressions that consist of variables with non-negative integer exponents. Polynomials can have multiple terms.
A simple polynomial like \(y^{2} + 2\) includes:
  • Terms: Each part of the expression separated by a plus or minus sign.
  • Exponent: In \(y^{2}\), the exponent is 2, indicating the degree of this term.
  • Monomials: A single term like \(y^{2}\) or a constant, here 2.
Understanding polynomials is crucial for operations such as addition, subtraction, and using properties like distributive to simplify or alter expressions.
Simplifying Expressions
Simplifying expressions often involves removing parentheses and combining like terms using properties like the distributive property.
Let's break down this process:
  • Distributive Property: Helps distribute a number outside the parentheses to each term inside. In \(-3(y^{2}+2)\), \(-3\) multiplies both \(y^{2}\) and 2.
  • Combining Like Terms: Once the expression is expanded, combine similar terms, if any. Though \(-3y^{2}\) and \(-6\) are different types of terms, they form the simplified expression \(-3y^{2} - 6\).
Through these steps, complex expressions become more straightforward and manageable, making them easier to work with during problem-solving.

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