Problem 12 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$ \frac{7-m}{m^{2}-49} $$

Step-by-Step Solution

Answer

\(\frac{7-m}{m^{2}-49} = \frac{7-m}{(m+7)(m-7)}\), this is the simplest form of the expression

Step 1: Identify the difference of squares

The denominator of the fraction \(m^{2} - 49\) is a difference of squares. In comparison to the formula \(a^{2}-b^{2}=(a+b)(a-b)\), here \(a=m\) and \(b=7\).

Step 2: Factorize the denominator

Use the difference of squares formula to decompose \(m^{2}-49\), giving us \((m+7)(m-7)\).

Step 3: Write the simplification

Substitute the denominator into the original expression. Giving us \(\frac{7-m}{m^{2}-49} = \frac{7-m}{(m+7)(m-7)}\) which can't be further simplified. So, the expression is already in its simplest form.

Step 4: Final verification

Ensure you have simplified as much as possible. Since no numerators cancel out with the denominators, the expression is at its simplest.

Key Concepts

Difference of Squares

The difference of squares is a mathematical identity that allows us to simplify expressions in a very elegant way. It takes the form \(a^2 - b^2 = (a+b)(a-b)\). When you see two square terms with a subtraction between them, it's probably a difference of squares.In the given exercise, we identify the expression \(m^2 - 49\) as a difference of squares. Here, \(m^2\) and \(49\) are both squares because they can be written as \(m^2 - 7^2\). The beauty of this method lies in its simplicity – you instantly factorize the expression into two binomials.Identifying these patterns can greatly speed up the simplification process, and it's crucial when dealing with polynomial expressions. Keep an eye out for squared terms and subtraction; they are your cues for applying the difference of squares.

Factoring

Factoring is one of the essential techniques in algebra. It involves breaking down expressions into a product of simpler expressions or factors. When we factor an expression, we express it as a product of its factors, making it simpler or more manageable. Let's revisit \(m^2 - 49\). We use the difference of squares formula to factor it into \((m+7)(m-7)\).This technique not only helps in simplifying expressions, but it is also instrumental when solving equations, integrating, and more. Knowing different factoring techniques can bring you closer to finding solutions and make complex tasks feel easier. Remember, practice makes perfect, so keep practicing different factorization problems to build your skill.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both, contain algebraic expressions. Simplifying such fractions follows the same principles as with numerical fractions.You look for factors in the numerator and the denominator that you can cancel out. In our exercise, we had the expression \(\frac{7-m}{m^{2}-49}\) which was simplified to \(\frac{7-m}{(m+7)(m-7)}\). Despite the factorization in the denominator, the expression couldn't be simplified further because there's nothing common to cancel between 7-m and the factors \((m+7)(m-7)\).Understanding how to simplify algebraic fractions often requires factoring and recognizing patterns. It helps in not only simplifying expressions but also in solving equations where these fractions appear. Always be careful with simplifications, and ensure all conditions are met for the expression to remain valid.