Question

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{a b x-b x^{2}}{a c x-c x^{2}}$$

Short answer

Reduced to lowest terms, the expression simplifies to \(\frac{b}{c}\).

Step-by-step solution

Factor Out Common Terms in Numerator and Denominator

Identify and factor out the common terms in both the numerator and the denominator. In the numerator, factor out the common term bx. In the denominator, factor out the common term cx.

Write the Factored Expression

Express the factored terms separately. The numerator becomes bx(a - x) and the denominator becomes cx(a - x).

Cancel Common Factors

Cancel out the common factors in the numerator and the denominator which are bx and cx and also the common binomial factor (a - x) to simplify the fraction.

Write the Simplified Expression

After canceling the common factors, the expression simplifies to b/c.

Ensure No Negative Exponents

Verify that the simplified answer does not contain any negative exponents. In this case, there are no negative exponents.

Key concepts

Factoring Algebraic Expressions

Understanding how to factor algebraic expressions is crucial in simplifying complex fractions and equations. Factoring involves breaking down a complicated expression into simpler parts that, when multiplied together, give you the original expression.

Consider the expression \( abx - bx^2 \). Both terms include a \( bx \), making it the common factor. When you factor out \( bx \) from both terms, you get \( bx(a - x) \). Similarly, in the denominator \( acx - cx^2 \), \( cx \) is the common factor, resulting in \( cx(a - x) \).

This process simplifies the algebraic fraction and prepares it for further reduction, such as canceling out common factors across the numerator and denominator.

Canceling Common Factors in Fractions

Once you've factored the algebraic expressions in the numerator and the denominator, it's time to look for and cancel out the common factors. This is a critical step in simplifying fractions. In our example, the factored expressions \( bx(a - x) \) and \( cx(a - x) \), both share a \( (a - x) \) term and the variables b and c are in their simplest form.

By canceling the common \( (a - x) \) factor, we essentially divide both the numerator and the denominator by \( (a - x) \), leaving us with \( b/c \). This cancellation leads to a much simpler expression, making the complex fraction easier to work with and understand.

Writing Expressions Without Negative Exponents

The final step in simplifying algebraic fractions is to ensure that the resulting expression has no negative exponents. In mathematics, negative exponents denote division by that base to the positive exponent. However, it's standard practice to write expressions without them, for clarity and simplicity.

In our original problem, after factoring and canceling the common factors, we're left with \( b/c \), which has no exponents at all—let alone negative ones. It's still important to check, as overlooking a negative exponent can lead to incorrect interpretations of the expression. If you ever encounter a negative exponent, remember that \( x^{-n} = 1/x^n \) is a useful equivalence to re-write the expression without negative exponents.