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Chapter 4: Problem 97

Simplify. $$3\left(5 x^{2}+2 x\right)$$

Short Answer

Expert verified
The simplified form of the given expression is \(15x^2 + 6x\)

Step by step solution

01

Distribute 3 to each term within the parentheses

Start by distributing the \(3\) to each term within the parentheses. This results in: \( 3*5x^2 + 3*2x \)
02

Perform the Multiplication

Next, carry out the respective multiplication operations. This gives us: \( 15x^2 + 6x \)

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Key Concepts

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

Distributive Property
The distributive property is a fundamental rule in algebra that helps to simplify expressions and solve equations. It essentially involves spreading or distributing a single term across terms in parentheses. The property states that for any numbers or variables, a, b, and c, the expression \( a(b + c) \) is equivalent to \( ab + ac \).

To understand the concept better in the context of the given exercise, consider how the number 3 is distributed across each term in the parenthesis \( (5x^2 + 2x) \).
  • You first multiply \( 3 \) by \( 5x^2 \), which results in \( 15x^2 \).
  • Then, you multiply \( 3 \) by \( 2x \), resulting in \( 6x \).
Using the distributive property simplifies the process of tackling more complex polynomial expressions. It breaks down tasks into manageable steps.
Monomials
Monomials are a core element of algebraic expressions. They are simply products of numbers and variables raised to a non-negative integer power. A monomial can have one or more variables, but they must be multiplied together, not added or subtracted.

Here are examples of monomials from the exercise solution:
  • \( 5x^2 \) is a monomial because it consists of the number 5 and a variable \( x \) raised to the power of 2.
  • \( 2x \) is another monomial as it includes the coefficient 2 with the variable \( x \) raised to the power of 1.
When simplifying algebraic expressions, ensure you respect the rules of exponents and only combine like terms, which are terms that have the exact same variable raised to the same power.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations like addition, subtraction, multiplication, and division. They do not include an equality or inequality sign. In the exercise, the complete expression is \( 3(5x^2 + 2x) \).

Here鈥檚 what makes an algebraic expression useful:
  • They allow us to represent general relationships mathematically.
  • They are central to formulating equations and inequalities.
  • They give us a way to express real-world situations in mathematical terms.
The key to working with algebraic expressions like the one in our example is understanding the components and operations within them. Simplification often involves identifying like terms and applying properties such as the distributive property to make the expression more manageable and easier to analyze.

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