Question

Expressions that occur in calculus are given. Reduce each expression to lowest terms. $$ \frac{\left(x^{2}+9\right) \cdot 2-(2 x-5) \cdot 2 x}{\left(x^{2}+9\right)^{2}} $$

Short answer

The expression simplifies to \[ \frac{-2x^2 + 10x + 18}{(x^2 + 9)^2} \]

Step-by-step solution

Simplify the numerator

Expand and simplify the numerator \[ \frac{\big((x^{2} + 9) \cdot 2 - (2x - 5) \cdot 2x\big)}{(x^{2} + 9)^{2}} \] which becomes: \[ \frac{2(x^2 + 9) - 2x(2x - 5)}{(x^2 + 9)^{2}} \]

Distribute terms

Distribute the 2 across \(x^2 + 9\) and \(2x\) across \(2x - 5\):\[ 2x^2 + 18 - 4x^2 + 10x \]

Combine like terms in the numerator

Combine the like terms resulting from distribution:\[ (2x^2 - 4x^2 + 10x + 18) = (-2x^2 + 10x + 18) \] which simplifies to \[ -2x^2 + 10x + 18 \]

Write simplified numerator over the denominator

Place the simplified numerator over the denominator:\[ \frac{-2x^2 + 10x + 18}{(x^2 + 9)^2} \]

Key concepts

algebraic fractions

When dealing with algebraic fractions, it is important to understand how to simplify them. Algebraic fractions are similar to numerical fractions, but they contain algebraic expressions instead of just numbers. They are in the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. Simplifying these fractions often involves factoring and reducing the numerator and the denominator, canceling any common factors if possible to achieve the simplest form.

numerator simplification

To simplify the numerator of an algebraic fraction, you need to manipulate the expression until it is as simplified as possible. In the exercise: \[ \frac{\big((x^{2} + 9) \cdot 2 - (2x - 5) \cdot 2x\big)}{(x^{2} + 9)^{2}} \], step 1 involves expanding and simplifying the numerator.

Begin by distributing terms within the numerator: \[ 2(x^2 + 9) - 2x(2x - 5) \], which simplifies to \[ 2x^2 + 18 - 4x^2 + 10x \].

Next, combine like terms to get: \[ -2x^2 + 10x + 18 \]. This is the simplified form of the numerator, ready to be placed over the denominator.

denominator handling

Handling the denominator in algebraic fractions is usually simpler, but it is also an important part of the process. In our example, the denominator \( (x^2 + 9)^2 \) remains unchanged through the simplification process of the numerator.

Often, you look out for factors that can be canceled out with terms in the numerator. However, in this exercise, the denominator is already in its simplest form. Ensure it is presented properly in the final reduced fraction, so our resulting simplified expression looks like \[ \frac{-2x^2 + 10x + 18}{(x^2 + 9)^2} \].

like terms

Like terms are terms within a polynomial that have the same variable raised to the same power. In the simplification process, combining like terms makes the expression more manageable. In our example:

Combining \( 2x^2 \) and \( -4x^2 \) results in \( -2x^2 \). Similarly, the term \(10x\) is simply added since there are no other \(x\) terms to combine it with. Terms like \(18\) are constants and remain untouched.

Recognizing and combining like terms is crucial as it reduces the complexity of algebraic expressions and makes further steps clearer. This leads us to the simplified numerator \( -2x^2 + 10x + 18 \), ready to be used in the final fraction.