Problem 13 - Elementary Algebra

Question

For the following problems, the first quantity represents the product and the second quantity a factor. Find the other factor. $$ 45 y^{2}+50 y, \quad 5 y $$

Step-by-Step Solution

Answer

Answer: The missing factor is \(4a - 1\).

Step 1: Write down the given information

We are given the following product and factor: Product: \(24a^2 - 6a\) Factor: \(6a\)

Step 2: Find the other factor

To find the missing factor, we will divide the product by the given factor: $$ \frac{24a^2 - 6a}{6a} $$

Step 3: Simplify the expression

Now let's simplify the expression by dividing each term by \(6a\): $$ \frac{24a^2}{6a} - \frac{6a}{6a} $$ $$ 4a - 1 $$ Hence, the other factor is \(4a - 1\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. In algebra, polynomials are expressions that consist of variables and coefficients.
When dividing polynomials, the division is executed similarly to the long division of numbers.

• We identify the dividend, which is the polynomial to be divided.
• The divisor is the polynomial by which we divide.

In the context of our problem, the dividend is the product given, \(24a^2 - 6a\), and the divisor is \(6a\).
The goal is to find the quotient, the missing factor.First, each term in the dividend is divided individually by the divisor.
This process simplifies the polynomial by reducing the degree of each term step by step.
It's crucial to carefully handle the variable powers and coefficients during this process to ensure correctness.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This means performing operations such as combining like terms, factoring, and reducing fractions.
For instance, in the given task:

• The product \(24a^2 - 6a\) was simplified by dividing each term separately by \(6a\).
• Dividing \(24a^2\) by \(6a\) involves dividing the coefficients (\(24 \div 6\)) and subtracting the exponents of like bases \(a^{2-1}\) resulting in \(4a\).
• Similarly, dividing \(6a\) by \(6a\) simplifies to \(1\) as any non-zero number divided by itself equals \(1\).

By performing these operations, we obtain a simpler expression, in this case, \(4a - 1\).
Simplifying expressions allows us to work with cleaner, more manageable forms of the polynomials, making further operations easier.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. Simplifying these fractions is pivotal to solving many algebraic problems.
In polynomial division, algebraic fractions emerge when dividing terms individually.
Take the expression \[\frac{24a^2 - 6a}{6a}\], for example. To simplify:

• We divide each term of the numerator by the denominator separately.
• For \(\frac{24a^2}{6a}\), we simplify by dividing the coefficients (24 by 6) and subtracting the exponents of \(a\), resulting in \(4a\).
• For \(\frac{6a}{6a}\), the entire fraction simplifies to \(1\) because dividing a term by itself results in \(1\).

As a result, the algebraic fraction \[\frac{24a^2 - 6a}{6a}\] simplifies to the algebraic expression \(4a - 1\).
Simplifying algebraic fractions is a key skill in algebra that aids in solving more complex equations.