Problem 84 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{\left(x^{2}-1\right)^{4}(2 x)-x^{2}(4)\left(x^{2}-1\right)^{3}(2 x)}{\left(x^{2}-1\right)^{8}}$$

Step-by-Step Solution

Answer

\( \frac{-6x^3 - 2x}{(x^2 - 1)^5} \) is the simplified expression.

Step 1: Factor Out Common Terms

In the given expression, notice that both terms in the numerator have common factors. Both terms include \((x^2 - 1)^3\) and \(2x\). Factor these out: \[\frac{(x^2 - 1)^3 (2x) \left[ (x^2 - 1) - 4x^2 \right]}{(x^2 - 1)^8}\]

Step 2: Simplify the Remaining Expression Inside the Brackets

Now simplify the expression inside the brackets:\[(x^2 - 1) - 4x^2 = x^2 - 1 - 4x^2 = -3x^2 - 1\]So, the expression inside the brackets becomes \(-3x^2 - 1\).

Step 3: Substitute the Simplified Expression Back

Substitute the simplified form back into the fraction:\[\frac{(x^2 - 1)^3 (2x) (-3x^2 - 1)}{(x^2 - 1)^8}\]

Step 4: Simplify the Fraction

Cancel the common term \((x^2 - 1)^3\) in the numerator and denominator:\[\frac{2x(-3x^2 - 1)}{(x^2 - 1)^5}\]

Step 5: Distribute in the Numerator

Distribute the \(2x\) in the numerator:\[\frac{-6x^3 - 2x}{(x^2 - 1)^5}\]

Step 6: Final Answer

The final simplified form of the expression is:\[\frac{-6x^3 - 2x}{(x^2 - 1)^5}\]

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying expressions. When you factor a polynomial, you're breaking it down into simpler pieces, called factors, that multiply together to give you the original polynomial. These factors are often easier to work with, especially when simplifying complex expressions. In this example,

• we identified that the terms in the numerator both share the factor \( (x^2 - 1)^3 \) and \( 2x \).
• Factoring allows us to rewrite the expression so that common terms can be canceled later on in the simplification process.

Recognizing common factors is the first step to simplifying rational expressions, and it's a critical skill in algebra. By practicing this method, you will be able to simplify and solve algebraic problems more efficiently.It's also important to remember that factoring isn't always immediately obvious. It requires practice and sometimes creative insight to see how expressions can break apart into their constituent pieces.

Simplifying Rational Expressions

Once you've factored polynomials, you can use simplification techniques to reduce the expression to its simplest form. A rational expression is simply a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions often involves canceling common factors between the numerator and the denominator.In our exercise,

• we factored out common terms from the numerator to simplify the expression inside the brackets.
• This involved performing basic algebraic operations like subtraction \(x^2 - 1 - 4x^2 \) to reduce the expression to \(-3x^2 - 1\).
• After substituting this simplified result back, we canceled common terms further, which greatly reduced the complexity of the fraction.
• Ultimately, this led us to a more manageable expression.

Simplifying rational expressions not only makes them easier to work with but also helps reveal essential features of the expression, such as its domain and potential undefined points, often critical in calculus and higher math.

Polynomial Division

Polynomial division can be thought of as similar to dividing numbers, but it deals with expressions made up of variables. In many cases, like in the given exercise, you perform polynomial division in the form of canceling common terms.Here:

• Once common terms \( (x^2 - 1)^3 \) were canceled from both the numerator and the denominator, we effectively "divided" the expression.
• After canceling, the fraction simplified from a complex polynomial to one that was much more straightforward.
• This simplification involves knowing how to recognize factors appropriately and understanding how they behave within fractions.

Understanding polynomial division is essential for managing expressions in algebra. It enables the transformation of an unwieldy expression into a simpler one that's often easier to analyze or use in subsequent calculations.