Problem 59 - Elementary Algebra

Question

Given that \(3 m^{2} n\) is a factor of \(12 m^{3} n^{4}\), find the other factor.

Step-by-Step Solution

Answer

Answer: The other factor is \(4mn^3\).

Step 1: Divide the expressions

First, we need to divide the expression \(12 m^3 n^4\) by the factor \(3 m^2 n\). We can do this by dividing each term in the expression by the corresponding term in the factor: $$\frac{12 m^3 n^4}{3 m^2 n} = \frac{12}{3} \times \frac{m^3}{m^2} \times \frac{n^4}{n}$$

Step 2: Simplify the fractions

Now, we simplify the fractions formed during the division: $$\frac{12}{3} \times \frac{m^3}{m^2} \times \frac{n^4}{n} = 4\times m^{3-2}\times n^{4-1}$$

Step 3: Combine the simplified terms

Finally, we combine the simplified terms to find the other factor: $$4\times m^{3-2}\times n^{4-1} = 4mn^3$$ Therefore, the other factor is \(4mn^3\).

Key Concepts

Factorization

Factorization is a fundamental concept in algebra where we express a number or expression as a product of its factors. When we talk about factorization in the context of algebraic expressions, such as polynomials, we aim to break down the expression into simpler expressions that when multiplied together give the original expression.

• This is very useful because it simplifies expressions and equations.
• It helps in finding roots or solutions to algebraic equations.
• In our example, we are required to find a missing factor such that when multiplied by the given factor, it results in the original polynomial.

To achieve this, we perform division of the original expression by the known factor. In this case, divide \(12m^3n^4\) by \(3m^2n\) to find the missing factor. We apply basic division rules by separating the constants and variables and simplifying each as much as possible. This simplifies the expressions and leads us to the solution, which shows the product's other factor.

Polynomials

Polynomials are expressions consisting of variables, also known as indeterminates, and coefficients, that involve addition, subtraction, multiplication, or non-negative integer exponents of variables. A key property of polynomials is that they can be easily factored and simplified to make calculations easier.

• They appear in various forms, such as monomials (single term), binomials (two terms), and trinomials (three terms).
• The expression \(12m^3n^4\) is a polynomial because it contains more than one term with variables raised to various powers.

When dealing with polynomials, we often want to simplify or rearrange them to reveal important features, like factors or roots. This requires operations like division, multiplication, and factorization. These methods help us understand the behavior of the polynomial better and solve equations more effectively. By dividing one polynomial by another, as in our example, we can find potential factors of a larger polynomial.

Exponents

Exponents are part of the algebra language that indicate how many times a number or variable is multiplied by itself. An exponent is written as a small number to the right and above a base number. Understanding how exponents work is essential in manipulating algebraic expressions.

• Exponents allow us to write repeated multiplication compactly, e.g., \(m^3\) represents \(m \times m \times m\).
• Basic properties include the power rules: for instance, \(a^m \times a^n = a^{m+n}\).
• In division, exponents are subtracted, such as \(\frac{m^3}{m^2} = m^{3-2} = m^1 = m\).

In the given example, understanding exponents allowed us to simplify the division of \(12m^3n^4\) by \(3m^2n\). We subtracted the exponents for like bases and simplified the result to find the remaining factor. Mastering exponents is crucial for simplifying polynomials and dealing effectively with variables in algebra.