Problem 25 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$\frac{7 x}{12 x+x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{7}{12+x}\).

Step 1: Identify the terms

In the given expression \(\frac{7x}{12x+x^{2}}\), 7x is the numerator and \(12x + x^{2}\) is the denominator.

Step 2: Factorise the denominator

The first step in simplifying a fraction is to factorise the numerator and the denominator if it's possible. In our case the denominator \(12x+x^2\) can be factorised by taking x common. So we get \(x(12 + x)\) as the factored denominator.

Step 3: Cancel out similar terms

Now we compare the numerator and the denominator for any similar terms that can be cancelled out. We observe that 'x' is common in both the numerator and the denominator, so we cancel out 'x'. Hence, the simplified fraction is \(\frac{7}{12+x}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler 'factor' components that, when multiplied together, produce the original polynomial. It's like taking a number apart to find numbers that multiply to make it. This is a crucial first step when simplifying algebraic expressions and solving equations.

Consider a polynomial like \( x^2 + 12x \), which appears in the denominator of our example fraction. To factor it, we look for common factors among the terms. In this case, both terms have an \( x \), so we factor it out, resulting in \( x(12 + x) \). Essentially, we're reversing the distributive property. This simplified form is highly useful for further operations, such as simplifying algebraic fractions.

Canceling Common Terms

Once a polynomial has been factored, simplifying the algebraic fraction often involves canceling common terms from the numerator and denominator—much like reducing a numerical fraction. Canceling common terms means dividing both the numerator and the denominator by the same nonzero quantity.

In our example, \( \frac{7x}{x(12 + x)} \), 'x' is a common term in both the numerator and the factorized denominator. Because division by a nonzero term is allowed, we can 'cancel' the common 'x' terms, which is equivalent to dividing both numerator and denominator by 'x'. What remains is the simplified expression \( \frac{7}{12 + x} \), free of the common factor. Remember that you can only cancel factors (terms that are multiplied), not terms that are added or subtracted.

Algebraic Expression Simplification

Simplifying algebraic expressions is about making expressions as easy to understand as possible. It's about removing complexities without changing the value. After factoring and canceling, you're often left with a more manageable expression. Simplification might involve combining like terms, reducing fractions, and canceling common factors.

The goal of algebraic expression simplification is to present the expression in its simplest form. This makes it easier to evaluate, differentiate, or integrate the expression, as well as to solve equations. The process highlighted in our original exercise—factoring the denominator, then canceling the common 'x'—transformed the complex fraction into a form that's much simpler to work with: \( \frac{7}{12 + x} \).