Question

Expressions that occur in calculus are given. Reduce each expression to lowest terms. $$ \frac{x \cdot 2 x-\left(x^{2}-4\right) \cdot 1}{\left(x^{2}-4\right)^{2}} $$

Short answer

\[\frac{x^2 + 4}{(x - 2)^2 (x + 2)^2}\]

Step-by-step solution

Simplify the Numerator

First, simplify the expression in the numerator: \[\frac{x \cdot 2x - (x^2 - 4) \cdot 1}{(x^2 - 4)^2}\]Perform the multiplication and distribute the minus sign: \[2x^2 - (x^2 - 4) = 2x^2 - x^2 + 4 = x^2 + 4\]

Replace the Numerator

Replace the numerator with its simplified form: \[\frac{x^2 + 4}{(x^2 - 4)^2}\]

Identify the Denominator

Notice that the denominator can be rewritten using the difference of squares: \[(x^2 - 4)^2 = (x^2 - 2^2)^2 = (x - 2)^2 (x + 2)^2\]

Final Expression

Combine the simplified numerator and the factored form of the denominator to form the simplest expression: \[\frac{x^2 + 4}{(x - 2)^2 (x + 2)^2}\]

Key concepts

Simplifying Fractions

Simplifying fractions involves reducing the numerator and the denominator to their smallest possible values without changing the original value of the fraction. It means making the numbers as simple as possible to work with. When you simplify, you find the greatest common divisor (GCD) of both the numerator and the denominator and divide both by this number.

In the given problem, the initial fraction was a complex expression. By performing operations such as multiplication and distributed subtraction, the numerator was simplified. Simplifying the fraction further means breaking down complex terms until they cannot be simplified anymore.

Remember, the goal is to make the fraction easier to understand and use. Always look for common factors in the numerator and the denominator and cancel them out to make the fraction simpler.

Difference of Squares

The difference of squares is a special factoring formula used to simplify expressions of the form \(a^2 - b^2\). The formula states that \(a^2 - b^2 = (a - b)(a + b)\).

This formula is very useful and frequently used in algebra and calculus.

For example, in the given expression, the denominator \(x^2 - 4\) can be rewritten as \(x^2 - 2^2\), which fits the form of a difference of squares. Applying the formula, it becomes \( (x - 2)(x + 2)\).

In our problem, this form was squared, so we further factored the expression to \((x - 2)^2 (x + 2)^2\). Understanding how to apply the difference of squares is crucial for simplifying and solving many algebraic expressions.

Factoring Polynomials

Factoring polynomials means breaking down a polynomial into simpler terms (factors) that, when multiplied together, give the original polynomial. This is a fundamental technique used to simplify expressions and solve polynomial equations.

Steps for factoring include:

• Looking for common factors in all terms.
• Applying special formulas like the difference of squares or the sum of cubes.
• Rewriting complex polynomials as product of simpler polynomials.

In the given exercise, we used factoring to simplify both the numerator and the denominator. The numerator \(2x^2 - (x^2 - 4)\) simplified to \(x^2 + 4\), and the denominator, using the difference of squares, factored to \((x - 2)^2 (x + 2)^2\). Understanding how to factor polynomials ensures you can reduce expressions to their simplest forms.

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It involves derivatives and integrals.

In calculus, simplifying rational expressions is critical as it allows you to work with more manageable forms of equations. This simplification helps in differentiating and integrating functions efficiently.

The given problem required reducing an expression to its lowest terms, a foundational skill in calculus. By simplifying and factoring the expression, we make it easier to apply calculus tools like limits, derivatives, and integrals.

Keep practicing simplifying and factoring to improve your calculus skills. These techniques are the building blocks for tackling more complex calculus problems.