Question

Expressions that occur in calculus are given. Reduce each expression to lowest terms. $$ \frac{\left(x^{2}+1\right) \cdot 3-(3 x+4) \cdot 2 x}{\left(x^{2}+1\right)^{2}} $$

Short answer

\( \frac{-3x^2 - 8x + 3}{(x^2 + 1)^2} \)

Step-by-step solution

- Expand the Numerator

First, expand and simplify the numerator: \( (x^2 + 1) \cdot 3 - (3x + 4) \cdot 2x \). This gives: \( 3x^2 + 3 - (6x^2 + 8x) \).

- Simplify the Numerator

Combine like terms in the numerator: \( 3x^2 + 3 - 6x^2 - 8x \). This simplifies to: \( -3x^2 - 8x + 3 \).

- Substitute Back into the Fraction

Insert the simplified numerator back into the original fraction: \( \frac{-3x^2 - 8x + 3}{(x^2 + 1)^2} \).

Key concepts

calculus expressions

When working with calculus expressions, it's crucial to know how to manipulate them to simplify or solve problems. Calculus expressions often involve functions, derivatives, and integrals. In our example, we deal with simplifying a fraction that involves polynomials. These skills are fundamental in more advanced calculus problems, such as finding limits, derivatives, and evaluating integrals.

expanding expressions

Expanding expressions is a critical step in simplifying complex algebraic problems. Here, we start by expanding the numerator. The original expression for the numerator is:

• \( (x^2 + 1) \times 3 - (3x + 4) \times 2x \)

To expand, distribute each multiplication across the terms inside the parentheses:

• \( 3(x^2 + 1) - 2x(3x + 4) \)
• \( 3x^2 + 3 - 6x^2 - 8x \)

This gives us the expanded form of the numerator.

simplifying fractions

Simplifying fractions involves reducing both the numerator and the denominator to their simplest forms. Once we expand the numerator, we combine any like terms. For instance, in our expression:

• Original expanded form: \( 3x^2 + 3 - 6x^2 - 8x \)
• Combined like terms: \( -3x^2 - 8x + 3 \)

After simplifying the numerator, we place it back over the original denominator:

\( \frac{-3x^2 - 8x + 3}{(x^2 + 1)^2} \).

combining like terms

Combining like terms is one of the most basic yet essential algebraic skills. Like terms are terms whose variable parts are the same. In our exercise, after expanding the expressions, we identify and combine like terms in the numerator:

• \( 3x^2 \) and \( -6x^2 \) combine to \( -3x^2 \)
• \( 3 \) remains as it is.
• \( -8x \) remains as it is.

This results in the simplified numerator: \( -3x^2 - 8x + 3 \). Combining like terms is foundational as it makes complex expressions easier to handle, understand, and compute.