Problem 18 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 8 a^{2}+4 a, 4 a $$

Step-by-Step Solution

Answer

Answer: The other factor is \(2a + 1\).

Step 1: Set up the division

We start by setting up the division as follows: $$ \frac{8a^2 + 4a}{4a} $$

Step 2: Divide the product by the factor

Next, we divide each term in the product by the factor: $$ \frac{8a^2}{4a} + \frac{4a}{4a} $$

Step 3: Simplify the division

Finally, we simplify each term in the division: $$ 2a + 1 $$ So, the other factor is \(2a + 1\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, similar to the way long division is used with numbers. This process is often employed to simplify expressions or find missing factors. In the given exercise, the goal is to determine the remaining factor when dividing a polynomial by a known factor.

To set up polynomial division, place the polynomial you want to divide as the 'numerator' and the divisor, which is the known factor, as the 'denominator'. For example, in our problem, we have
\[\frac{8a^2 + 4a}{4a}\]
This setup simulates the division process, allowing us to divide each term in the polynomial individually by the divisor.

• The numerator consists of the terms \(8a^2\) and \(4a\).
• The denominator is the factor \(4a\).

By understanding polynomial division, you can break down complex expressions into simpler parts.

Factoring

Factoring involves breaking down a complex expression into a product of simpler expressions. This is essential in solving algebraic equations, simplifying expressions, or finding roots. In our exercise, it helps identify the other factor when one factor is already known.

To factor an expression, you search for common terms that can be 'factored out'. In the polynomial \(8a^2 + 4a\), notice that \(4a\) can be factored out from both terms. This common factor is crucial as it forms part of the solution.

• Look for the greatest common factor (GCF) among the terms.
• Factor out the GCF.

In this case, the GCF is \(4a\), and when factored out, we can see that:
\[8a^2 + 4a = 4a(2a + 1)\]
This result shows the expression's factorized form, revealing the other factor \(2a + 1\).

Simplification

Simplification is the process of rewriting expressions in their simplest form. This makes them easier to understand and work with, which is especially important when solving equations.

In our provided solution, simplification occurs after dividing each term. Here, we take \(\frac{8a^2}{4a} + \frac{4a}{4a}\) and simplify each component:

• The term \(\frac{8a^2}{4a}\) simplifies to \(2a\).
• The term \(\frac{4a}{4a}\) simplifies to \(1\), since any number divided by itself is 1.

Combine these simplified components to find the simplest form of the expression, which in this case is:
\[2a + 1\]
Simplification leads us to clear and concise results, like identifying the other factor in multiplication problems. Mastering this skill is key in solving algebra problems effectively.