Problem 40 - Elementary Algebra

Question

For the following problems, fill in the missing term. $$ -\frac{x-2}{6 x-1}=\frac{\underline{\phantom{xx}}}{6 x-1} $$

Step-by-Step Solution

Answer

Answer: The missing term is (2m+1).

Step 1: Identify the common terms in the numerator and denominator

In the given expression, $$\frac{5 m^{2}}{10 m^{3}+5 m^{2}}$$, we can observe that both the numerator and the denominator have a common term of 5 and \(m^2\).

Step 2: Factor out the common terms from the numerator and denominator

Now, we will factor out the common terms, 5 and \(m^2\), from both the numerator and the denominator. After factoring 5 from the numerator: $$5m^2 = 5 \cdot m^2$$ After factoring \(m^2\) and 5 from the denominator: $$10m^3 + 5m^2 = 5m^2(2m + 1)$$

Step 3: Substitute the factored terms back into the expression

Now, replace the numerator and denominator in the original expression with their factored counterparts: $$\frac{5m^{2}}{10m^{3}+5m^{2}} = \frac{5 \cdot m^2}{5m^2(2m+1)}$$

Step 4: Simplify the expression

In this step, we will cancel the common terms in the numerator and the denominator: $$\frac{5 \cdot m^2}{5m^2(2m+1)} = \frac{1}{2m+1}$$ The missing term is (2m+1), and the simplified expression is: $$\frac{5m^2}{10m^3+5m^2} = \frac{1}{2m+1}$$

Key Concepts

Common Factors

Common factors are integral to simplifying algebraic expressions. They are the numbers or variables that can divide every part of an expression without leaving a remainder.
When you see an algebraic fraction, your first step is to identify the common factors in both the numerator and denominator.
The process begins by examining each term to find what they share.In our problem, we have an expression:

• Numerator: \(5 m^2\)
• Denominator: \(10 m^3 + 5 m^2\)

By carefully looking at these, it's apparent that both have the factors 5 and \(m^2\). By factoring these out, we simplify the expression and prepare it for further reduction.
This step is crucial as it allows us to remove redundancy and make calculations easier.

Simplifying Fractions

Simplifying fractions is a key skill in algebra, making expressions more manageable and revealing their core components. When you simplify a fraction, you're essentially dividing the numerator and the denominator by their greatest common factor (GCF).
This process is a fundamental activity in math that helps in creating equivalent but simpler expressions.After identifying common factors, as in our example, you factor them out:

• The numerator becomes \(5 \cdot m^2\).
• The denominator transforms into \(5m^2(2m + 1)\).

By cancelling the common \(5m^2\) in both the numerator and the denominator, we simplify our expression from \(\frac{5m^2}{10m^3+5m^2}\) to \(\frac{1}{2m+1}\).
Remember, the goal of simplification is to make the fraction easier to work with, without changing its value.

Polynomials

Polynomials are expressions that consist of variables and coefficients, with operations of addition, subtraction, multiplication, and non-negative integer exponentiation.
They can be simple, like just a single term \(m^2\), or more complex combinations like \(10m^3 + 5m^2\).Understanding polynomials is crucial because they form the backbone of algebra.
When looking at our polynomial – \(10m^3 + 5m^2\), it's key to recognize how it's composed and how factors interact within it.

• Each term in a polynomial is made up of a variable raised to an exponent and multiplied by a coefficient.
• Factoring polynomials involves breaking them down to their simplest building blocks, often using methods like finding common factors.

In this exercise, recognizing the polynomial nature of the denominator allowed us to factor it thoroughly, simplifying our original problem.