Problem 52 - Elementary Algebra

Question

Given that \(4 a^{3}\) is a factor of \(8 a^{3}-12 a^{2}\), find the other factor.

Step-by-Step Solution

Answer

Answer: The other factor is \(1 - \frac{3}{a}\).

Step 1: Write the given expression as a product.

We know that \(4 a^{3}\) is a factor of \(8 a^{3}-12 a^{2}\). Let's write that as a product: \(8 a^{3}-12 a^{2} = 4 a^{3} \times (\text{unknown factor})\)

Step 2: Solve for the unknown factor.

Now, to find the other factor, divide both sides of the equation we just wrote by \(4 a^{3}\): \((\text{unknown factor}) = \frac{8 a^{3}-12 a^{2}}{4a^3}\) Now, simplify by cancelling common factors: \((\text{unknown factor}) = \frac{8 a^{3} - 12 a^{2}}{4 a^{3}}\) =\((a^3 - 3a^2)\frac{4}{4}\) (factor 4 out of the numerator) =\((a^3 - 3a^2)\frac{1}{a^3}\) (divide out the \(4\)'s and factor out \(a^3\) from the numerator) \((\text{unknown factor}) = 1 - \frac{3}{a}\)

Step 3: State the other factor

We have found the other factor to be \(1 - \frac{3}{a}\). Thus, we can write the given expression as: \( 8 a^{3} - 12 a^{2} = 4 a^{3} \times \left(1-\frac{3}{a}\right)\)

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are like sentences in the language of mathematics. They are combinations of numbers, variables, and arithmetic operations. For instance, in the expression \(8a^3 - 12a^2\), \(8a^3\) and \(-12a^2\) are terms. Here, the numbers 8 and 12 are coefficients, and \(a\) is a variable raised to different powers.

In algebra, we often work with expressions to simplify, solve, and understand relationships between quantities. Recognizing and manipulating these components correctly is crucial for solving algebra problems, like identifying factors in an expression.

Division of Polynomials

Division of polynomials may sound complex, but it's similar to dividing numbers. It's the process of dividing one polynomial by another, typically aiming to simplify expressions or find factors.

Consider dividing \(8a^3 - 12a^2\) by \(4a^3\). The goal is to find an expression that when multiplied by \(4a^3\), gives back the original polynomial.

• First, divide each term of the numerator by \(4a^3\).
• Here, divide both \(8a^3\) and \(-12a^2\) separately by \(4a^3\).
• Simplify each term to find \(2 - \frac{3}{a}\).

Understanding this process can help simplify complex expressions and solve polynomial equations.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form without changing their value. It's like tidying up a messy equation, making it easier to work with or understand.

Take our exercise: \(\frac{8a^3 - 12a^2}{4a^3}\). We simplify by:

• Canceling out any common factors, such as the 4 and \(a^2\) from both the numerator and denominator.
• This leaves us with \(2a - 3\).

By simplifying, we identify that the unknown factor leading from the original expression \(8a^3 - 12a^2\) is actually \(2a - 3\). Simplification is often the final polishing step in algebra problems, revealing the simplest, cleanest form of an expression.