Problem 39 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ (-5 z)\left(2 z^{2}\right) $$

Step-by-Step Solution

Answer

The simplified version of the expression is \( -10z^3 \)

Step 1 - Identify terms to simplify

The expression given is \( (-5z)(2z^2) \). Here, \(-5z\) and \(2z^2\) are the terms that need to be multiplied.

Step 2 - Apply the distributive property of multiplication over addition

The distributive property can be applied as follows: \((-5z)\times (2z^2) = -5 \times 2 \times z \times z^2\). This simplifies to \( -10z^3 \).

Step 3 - Result

The simplified version of the algebraic expression \( (-5z)(2z^2) \) is \( -10z^3 \).

Key Concepts

Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify expressions and solve equations efficiently. When you encounter an expression like \(a(b + c)\), the distributive property tells us that you can 'distribute' the value of \(a\) to both \(b\) and \(c\), resulting in \(ab + ac\).

But what happens when we have no addition or subtraction inside the parentheses, as with the expression \( (-5z)(2z^2) \)? In this case, we distribute the multiplication across each term in the polynomials. We multiply the \( -5 \) and \( z \) by \( 2 \) and \( z^2 \) respectively. By applying this property systematically, we achieve our simplified result.

Understanding the distributive property is fundamental not just to simplifying algebraic expressions, but it also lays the groundwork for future topics such as factoring polynomials and solving complex equations.

Multiplying Polynomials

Multiplying polynomials might seem daunting at first, but with a systematic approach, it’s a breeze. To multiply polynomials, we take each term in the first polynomial and multiply it by every term in the second polynomial. In the given exercise, the polynomials are quite simple; one is a monomial, \( -5z \), and the other is also a monomial, \(2z^2 \).

When these terms are multiplied, as in \( (-5z)(2z^2) \), we multiply the coefficients (numeric parts) and the variables separately. For the coefficient part, \( -5 \times 2 = -10 \). For the variable part, we apply the exponent rule \(z^1 \times z^2 = z^{1+2} = z^3\), since when we multiply variables, we add their exponents. This process yields \( -10z^3 \), the simplified form of the expression.

A solid grasp of multiplying polynomials is instrumental in more complex operations, such as expanding binomials or finding the product of two or more polynomials, each encompassing several terms.

Algebraic Expressions

Algebraic expressions are the phrases of algebra—the building blocks that convey quantitative relationships with variables and constants combined using mathematical operations. An expression like \( (-5z)(2z^2) \) is a compact way of stating that we have \( -5 \) of something (in this case, \( z \)) and we're going to multiply that by \( 2z^2 \).

Understanding how to work with these expressions is crucial. It's not just about applying properties and rules; it's about recognizing patterns and predicting the form of the result. For instance, knowing that multiplying powers of the same base results in a new power with an exponent that is the sum of the original exponents can save time and confusion.

Moreover, the ability to manipulate these expressions is key to progressing in algebra and beyond. Whether it’s simplifying expressions, solving equations, or modeling real-world problems, algebraic expressions are the language we use to translate and solve a wide array of mathematical challenges.