Problem 4 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity represents a factor. Find the other factor. $$ 50 m^{3} n^{5} p^{4} q, \quad 10 m^{3} q $$

Step-by-Step Solution

Answer

Answer: The other factor is $$5 n^{5} p^{4}$$.

Step 1: Write down the given product and factor

The given product is $$50 m^{3} n^{5} p^{4} q$$ and the given factor is $$10 m^{3} q$$.

Step 2: Divide the product by the given factor

Divide $$50 m^{3} n^{5} p^{4} q$$ by $$10 m^{3} q$$, which can be written as: $$ \frac{50 m^{3} n^{5} p^{4} q}{10 m^{3} q} $$

Step 3: Simplify the result

Simplify the expression by canceling out the common terms between the numerator and the denominator and simplifying the coefficients: $$ \frac{50 m^{3} n^{5} p^{4} q}{10 m^{3} q} = \frac{50}{10} \cdot \frac{m^{3}}{m^{3}} \cdot n^{5} \cdot \frac{p^{4}}{1} \cdot \frac{q}{q} $$

Step 4: Calculate the other factor

Calculate the result from the expression obtained in Step 3: $$ 5 \cdot 1 \cdot n^{5} \cdot p^{4} \cdot 1 = 5 n^{5} p^{4} $$ Thus, the other factor is $$5 n^{5} p^{4}$$.

Key Concepts

Dividing Polynomials

When we are faced with a problem involving dividing polynomials, our goal is to separate one algebraic expression into its individual parts or factors. In essence, dividing polynomials is a process of simplification where we seek to express a complex algebraic expression as a quotient of two simpler expressions.

Imagine you are splitting a pizza into equal parts, where each slice represents a 'factor' of the pizza. Similarly, with algebra, dividing polynomials means we are finding parts of an expression that, when multiplied together, would give us the original expression.

For example, if we have a polynomial like \( 50 m^{3} n^{5} p^{4} q \) and we know that \( 10 m^{3} q \) is a factor of that polynomial, we can divide the two expressions to find the missing factor. We write it as a fraction, \[ \frac{50 m^{3} n^{5} p^{4} q}{10 m^{3} q} \], and then cancel out like terms that appear in both the numerator and the denominator to simplify.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is like cleaning up a cluttered room so everything is easier to find. The goal is to reduce the expression to its most basic form without changing its value.

We often start by combining like terms—those terms that have the same variables raised to the same power. We also simplify coefficients, which are the numerical parts of terms. It's essential to remember that only like terms can be combined; terms with different variables or exponents remain separate.

In our polynomial division example, the process of simplifying involves cancelling out the terms \(m^{3}\) and \(q\) because they appear in both the numerator and the denominator. We also simplify the coefficients by dividing \(50\) by \(10\), getting the simpler expression \(5 n^{5} p^{4}\), which represents the missing factor.

Exponentiation in Algebra

Exponentiation, the process of raising numbers to powers, is a fundamental concept in algebra that involves repeated multiplication. In our context, when we talk about exponentiation in algebra, we're usually dealing with variables raised to certain powers.

For instance, \(m^{3}\) means \(m\) is multiplied by itself three times. What's interesting in the context of dividing polynomials is that when we have the same base, like \(m\), raised to a power in both the numerator and the denominator, they cancel each other out—this is due to the rules of exponentiation.

Understanding how exponentiation works allows us to simplify expressions quickly. For example, \( \frac{m^{3}}{m^{3}} \) simplifies to \(1\), because any number (or variable) raised to the zeroth power is equal to one. This principle is crucial when dividing and simplifying algebraic expressions with exponents.