Problem 20 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ 39 x^{4} y^{5} z^{11}, \quad 3 x y^{3} z^{10} $$

Step-by-Step Solution

Answer

Answer: The other factor is \(13 x^{3} y^{2} z\).

Step 1: Identify the product and the given factor

The product we are given is \(39 x^{4} y^{5} z^{11}\), and the given factor is \(3 x y^{3} z^{10}\).

Step 2: Divide the product by the given factor

We now need to divide the product by the given factor to find the other factor. We will set up the division as follows: $$ \frac{39 x^{4} y^{5} z^{11}}{3 x y^{3} z^{10}} $$

Step 3: Divide the coefficients and the variables with the exponent rules

To perform the division, we can divide the coefficients (the numbers) and then use the exponent rules to divide the variables. We have: $$ \frac{39}{3} = 13 $$ As for the variables in the expression, we subtract the exponents when dividing: $$ \frac{x^{4}}{x} = x^{4-1} = x^{3} $$ $$ \frac{y^{5}}{y^{3}} = y^{5-3} = y^{2} $$ $$ \frac{z^{11}}{z^{10}} = z^{11-10} = z $$

Step 4: Combine the results

We now multiply the quotient obtained earlier to get the other factor: $$ 13 \cdot x^{3} \cdot y^{2} \cdot z $$ So, the other factor is: $$ 13 x^{3} y^{2} z $$

Key Concepts

Exponent Rules

Exponent rules are crucial in manipulating expressions involving powers. When dividing variables with exponents, you subtract the exponent of the denominator from the exponent of the numerator. This is known as the quotient rule for exponents. For instance, to simplify \( \frac{x^4}{x} \), you subtract the exponents as follows: \( 4 - 1 = 3 \). Similarly, \( \frac{y^5}{y^3} \) is simplified to \( y^{5-3} = y^2 \).

• Ensure both the base and the exponent are present before applying the rule.
• The subtraction only applies to like bases, i.e., the same variable.

Practicing these rules helps in simplifying complex algebraic expressions quickly.

Polynomial Division

Polynomial division involves breaking down a polynomial expression similar to how you would perform long division with numbers. Here, our task is to find the missing factor by dividing a given product by a known factor.

• Start by dividing the coefficients of the terms: \( \frac{39}{3} = 13 \).
• Next, apply the exponent rules to the variables, as explained in the previous section.

This strategic division allows you to isolate different components of a polynomial, making it simpler to solve for unknowns and understand relationships within an equation.

Variable Manipulation

Variable manipulation involves managing variables using algebraic techniques to simplify or solve expressions. Here, we're dealing with simplifying products by comparison and division. This involves:

• Recognizing like terms, in this case, the variables \( x, y, \) and \( z \).
• Organizing terms in a way that aligns them for division or other operations.

Align the variables vertically by their type, then subtract the exponents as needed. This step was crucial to finding the other factor in our exercise: \( 13x^3y^2z \).

Proficiency in variable manipulation enhances problem-solving skills in more complex algebraic tasks, paving the way for mastery in diverse mathematical applications.