Problem 8 - Algebra for College Students

Question

Find each product. $$\left(-2 x y^{2} z^{2}\right)\left(-x^{2} y^{3} z\right)$$

Step-by-Step Solution

Answer

The product is \(2x^3y^5z^3\).

Step 1: Identify and Multiply the Coefficients

The first expression is \(-2\), and the second expression does not have an explicit coefficient, meaning its coefficient is \(-1\). When multiplying, \((-2) imes (-1) = 2\). This gives the coefficient for the final product.

Step 2: Multiply the x Variables

The first expression has \(x\) and the second has \(x^2\). When multiplying these, use the property of exponents: \(x^1 imes x^2 = x^{1+2} = x^3\).

Step 3: Multiply the y Variables

In the first expression, \(y^2\) is present, and in the second expression, \(y^3\) is present. Multiply these together: \(y^2 imes y^3 = y^{2+3} = y^5\).

Step 4: Multiply the z Variables

In the first expression, \(z^2\) is present, and in the second expression, \(z\) is present. Multiply these together: \(z^2 imes z^1 = z^{2+1} = z^3\).

Step 5: Write the Final Product

Combine the results from Steps 1, 2, 3, and 4 to write the final expression: \(2x^3y^5z^3\).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, formed using addition, subtraction, multiplication, and non-negative integer exponents of variables. In the context of the exercise above, we are dealing with a type of polynomial known as a monomial. A monomial is simply a polynomial with a single term. The expression

• \(-2xy^2z^2\)
• \(-x^2y^3z\)

are examples of monomials.
Polynomials can range from simple, such as \(x + y\), to complex, involving multiple variables and terms, such as \(3x^2 + 2xy - y^2 + 7\). Working with polynomials requires understanding how to manipulate these terms, especially through multiplication and addition.
In the case of the exercise, multiplying two monomials involves multiplying the coefficients and then applying the properties of exponents to the variables involved. Understanding how to combine these elements is crucial in mastering algebra.

Multiplication of Variables

When multiplying variables within polynomials, each variable must be multiplied separately, taking care of their individual exponents. In the given exercise, the expressions

• \(-2xy^2z^2\)
• \(-x^2y^3z\)

are full of variables that need to be handled according to their respective rules.
Let's break it down:

• The variable \(x\): Multiply \(x\) by \(x^2\), which results in \(x^{1+2} = x^3\).
• The variable \(y\): Multiply \(y^2\) by \(y^3\), giving \(y^{2+3} = y^5\).
• The variable \(z\): Multiply \(z^2\) by \(z\), resulting in \(z^{2+1} = z^3\).

It's important to separate the coefficients from the variable terms when doing these calculations. The process involves adding exponents when variables are the same, which will be explained more under the properties of exponents.

Properties of Exponents

The properties of exponents are crucial when simplifying expressions with variables in algebra. They describe how to manage and manipulate powers of numbers and are vital in polynomial multiplication. In the given problem, understanding these properties helps simplify the multiplication of variables effectively.
Key properties include:

• Product of Powers: When multiplying two terms with the same base, add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising a power to another power, multiply the exponents, like \((a^m)^n = a^{m \cdot n}\). This property isn't directly involved in the exercise but is important in broader contexts.
• Power of a Product: Distribute the exponent to each factor within a product, such as \((ab)^n = a^n \cdot b^n\).

In this exercise, recognizing the product of powers is essential. It allows us to easily multiply variables \(x, y, z\) by adding their exponents together, resulting in a simplified form. This helps in creating the final expression, which in this instance is \(2x^3y^5z^3\). Mastering these properties makes handling complex expressions more manageable.