Problem 1 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity represents a factor. Find the other factor. $$ 32 a^{4} b, \quad 2 b $$

Step-by-Step Solution

Answer

Answer: The missing factor is $$16 a^{4}$$.

Step 1: Identify the given product and factor

The given product is $$32 a^{4} b$$, and the given factor is $$2 b$$.

Step 2: Apply the property of multiplication to find the missing factor

Using the property of multiplication, we can say that the missing factor (a) is equal to the product divided by the given factor: $$a = \frac{32 a^{4} b}{2 b}$$

Step 3: Divide the product by the given factor

Now, we will divide the product by the given factor: $$a = \frac{32 a^{4} b}{2 b} = 16 a^{4}$$ The missing factor is $$16 a^{4}$$.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are the backbone of algebra and are a combination of numbers, variables, and operators (such as +, -, *, and /) without an equality sign. For example, in the problem \( 32 a^{4} b \), \(32 a^{4} b\) is an algebraic expression where \(32\) is a coefficient, \(a^{4}\) is a term with \(a\) as the variable raised to the power of 4, and \(b\) is another variable.

To simplify algebraic expressions, we can perform operations such as addition, subtraction, multiplication, and division. When dividing algebraic expressions, as seen in our problem, understanding the terms and the relationship between them is crucial. Always look for common factors and apply the laws of exponents when dealing with variables.

Division of Polynomials

The division of polynomials involves dividing one polynomial by another. It's like standard long division but with variables. For the exercise given, we aren't dividing two polynomials but rather a monomial by another monomial. A monomial is a single term that is a product of numbers and variables raised to powers. Here, \(32 a^{4} b\) is the dividend, and \(2 b\) is the divisor.

In our step by step solution, we divided \(32 a^{4} b\) by \(2 b\) to find the other factor. To do this, we look for any common numerical coefficients and variables. Here, \(2\) divides into \(32\), giving us \(16\), and \(b\) cancels out since it is present in both the numerator and the denominator. The remaining expression, \(16 a^{4}\), is our desired factor. This process of simplifying helps in solving more complex polynomial division problems as well.

Multiplication Properties

Multiplication properties are rules that help us simplify and manage multiplication operations involving numbers and variables. These properties include the Commutative Property, which states that the order of factors does not affect the product; the Associative Property, which explains that the grouping of factors will not affect the product; and the Distributive Property, which allows us to multiply a sum by a number by multiplying each addend separately and then adding the products.

For our exercise, we applied a basic property of multiplication that states if a product is divided by one of its factors, the quotient is the remaining factor. From \(32 a^{4} b / 2 b = 16 a^{4}\), we see that dividing the product by the given factor gives us the missing factor, showcasing the inverse relationship between multiplication and division. Understanding these properties is essential for manipulating and simplifying algebraic expressions and equations.