Problem 25 - 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. $$ 5 a^{4} b^{7} c^{3} d^{2}, 5 a^{4} b^{7} c^{3} d $$

Step-by-Step Solution

Answer

Answer: \(d\)

Step 1: Identify the product and the given factor

In this exercise, the product is \(5a^{4}b^{7}c^{3}d^{2}\), and the given factor is \(5a^{4}b^{7}c^{3}d\).

Step 2: Divide the product by the given factor

Now, we need to divide the product by the given factor to find the other factor. Divide \(5a^{4}b^{7}c^{3}d^{2}\) by \(5a^{4}b^{7}c^{3}d\): $$\frac{5a^{4}b^{7}c^{3}d^{2}}{5a^{4}b^{7}c^{3}d}$$

Step 3: Simplify the expression

Now, we will simplify the expression by canceling out terms that are present in both the numerator and the denominator: $$\frac{5a^{4}b^{7}c^{3}d^{2}}{5a^{4}b^{7}c^{3}d} = \frac{d^{2}}{d}$$

Step 4: Find the other factor

Finally, we can find the other factor by dividing the remaining terms: $$\frac{d^{2}}{d}=d$$ The other factor is \(d\).

Key Concepts

Factoring

Factoring in algebra involves breaking down an expression into its simplest parts, known as factors. Think of factors as building blocks that are multiplied together to form the original expression. For example, in the exercise given, the product is expressed as a multiplication of variables and coefficients:

• The original product is: \(5a^{4}b^{7}c^{3}d^{2}\)
• The given factor is: \(5a^{4}b^{7}c^{3}d\)

To identify the other factor, we factor the given element out of the product. Understanding the relationship between factors helps in recognizing that every product consists of multiple factors. By dividing these expressions, you determine which multiplier completes the original factorization, in this case, the missing factor is the simplest form of the excess part of the product that was not part of the given factor.

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that makes equations more manageable and easier to work with. It involves reducing expressions to their simplest form by performing operations like canceling terms or factors that appear in both the numerator and the denominator.
In the solution, we simplified the initial complex expression \(\frac{5a^{4}b^{7}c^{3}d^{2}}{5a^{4}b^{7}c^{3}d}\). By canceling identical terms, you reduce the expression complexity, leaving only the necessary components. For our problem:

• Cancel \(5a^{4}b^{7}c^{3}\) in both numerator and denominator since they are identical.
• The expression simplifies to \(\frac{d^{2}}{d}\), where \(d\) can again be reduced, leaving the final simplified factor of \(d\).

Simplification makes expressions easier to understand and solve, forming the basis for solving equations and factoring polynomials efficiently.

Polynomials

Polynomials are expressions made up of variables raised to whole number powers and coefficients. They play a fundamental role in algebra as they represent a wide range of functions and can model various real-world scenarios. In our exercise:

• The original polynomial is \(5a^{4}b^{7}c^{3}d^{2}\).
• It consists of terms where each variable is a part of a power product expression with a numeric coefficient.

Understanding how to manipulate and work with polynomials is essential for tasks like factoring or simplifying algebraic expressions. When breaking down these expressions through factorization, as done in this exercise, you're often simplifying a polynomial into more workable parts.
Throughout different problems, you may encounter quadratic polynomials (like \(ax^2 + bx + c\)) or more complex structures, but the principles of evaluating, simplifying, and factoring remain consistently applicable.