Problem 23 - Elementary Algebra

Question

For the following problems, fill in the missing term. $$ \frac{5 m+25}{10 m^{2}+15 m} $$

Step-by-Step Solution

Answer

Answer: The missing term is 5. The simplified fraction is \(\frac{m+5}{m(2m+3)}\).

Step 1: Identifying common factor(s)

We need to find common factors between \(5m+25\) and \(10m^2+15m\). For this, let's factor out the greatest common factor from each expression: $$ \frac{5(m+5)}{5m(2m+3)} $$ Here, we can see that both numerator and denominator have a common factor of 5.

Step 2: Simplifying the fraction

Since we found a common factor of 5 in both the numerator and the denominator, we can simplify the fraction by dividing both the numerator and the denominator by 5: $$ \frac{5(m+5)}{5m(2m+3)} \div \frac{5}{5} $$ Which gives us the simplified fraction: $$ \frac{m+5}{m(2m+3)} $$ Now the missing term is found and the fraction is simplified.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial skill when working with algebraic expressions. It involves breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, give back the original polynomial. Imagine you are given a numerical expression like 6 * 10. You know that this is equal to 60, but what if you wanted to break down 60 into its components that are easier to work with? Factoring does exactly that, but with polynomials.

For example, if we have a polynomial like \(10m^2+15m\), we want to express it as a product of its factors. A common technique is to look for the greatest common factor, which is the highest number that divides all the terms. Here, both terms are multiples of 5m. By factoring out 5m, we get \(5m(2m+3)\), which is easier to manage, especially when simplifying algebraic fractions.

Common Factors

Identifying common factors is like finding what ingredients are shared in two different recipes so you can use them more efficiently or even eliminate the redundant parts. When we have two algebraic expressions, finding common factors helps us reduce the expression to its simplest form.

Take the terms \(5m+25\) and \(10m^2+15m\) from earlier. By breaking down each term, we identify that the number 5 is a factor in all parts of both expressions. Recognizing this, we can 'take out' the common factor, simplifying our work with the remaining unique parts of each expression. This not only makes the algebra easier but it's also essential when simplifying fractions because it can lead to canceling out the same factor from the numerator and denominator, as we saw with the factor of 5.

Simplifying Expressions

Simplifying expressions is the process of condensing an algebraic expression into its simplest form. It's similar to cleaning up your work desk; you're removing unnecessary items and organizing what's left in a more efficient manner. In algebra, this means combining like terms, canceling out common factors, and reducing fractions to their lowest terms.

Once we factored out the greatest common factor in the exercise, we were left with \(\frac{(m+5)}{m(2m+3)}\). This fraction is considered simplified because there are no more common factors that can divide both the numerator and the denominator evenly. It's in its most straightforward form, allowing for more effortless comprehension and further manipulation if required. Simplifying expressions helps to not only make the numbers and terms more manageable but also lays a strong foundation for solving more complex algebraic problems.