Question

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

Short answer

\[\frac{x(3x + 2)}{(3x + 1)^2}\]

Step-by-step solution

- Expand the Numerator

First, expand the numerator. The given expression is:\[\frac{(3x + 1) \times 2x - x^2 \times 3}{(3x + 1)^2}\]Expanding the numerator, we get:\[\frac{(6x^2 + 2x - 3x^2)}{(3x + 1)^2}\]

- Combine Like Terms in the Numerator

Next, combine the like terms in the expanded numerator:\[6x^2 + 2x - 3x^2 = 3x^2 + 2x\]So, the expression simplifies to:\[\frac{3x^2 + 2x}{(3x + 1)^2}\]

Ste 3 - Factor the Numerator

Factor the numerator to see if it can be simplified further:\[3x^2 + 2x = x(3x + 2)\]Thus, we rewrite the expression as:\[\frac{x(3x + 2)}{(3x + 1)^2}\]

- Simplify the Expression

Observe that the numerator and the denominator do not share any common factors other than 1, so the expression:\[\frac{x(3x + 2)}{(3x + 1)^2}\]is already in the lowest terms.

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes variables, numbers, and operations like addition, subtraction, multiplication, and division. For example, in our exercise, the expression \((3x + 1) \cdot 2x - x^2 \cdot 3\) is an algebraic expression. It represents a combination of variables and constants connected by arithmetic operations.
When simplifying algebraic expressions, you often need to perform several steps, which may include expanding, combining like terms, and factoring. Understanding how to manipulate these expressions is essential for solving algebra problems effectively.

Calculus Preparation

Simplifying rational expressions is an important skill for calculus preparation. In calculus, you frequently encounter complex expressions that need to be simplified to make them easier to work with.
In our exercise, simplifying the expression before working on it further can save a lot of time. Calculus often requires dealing with limits, derivatives, and integrals of functions. These operations become much more manageable when the underlying expressions are in their lowest terms.
Grasping basic algebraic manipulation helps build a strong foundation for more advanced topics in calculus.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as the product of its factors. For instance, in the given exercise, the numerator \(3x^2 + 2x\) is factored into \((x(3x + 2))\).
Here, the first step is recognizing any common factors in the terms of the polynomial. We see that both terms, \(3x^2\) and \(2x\), have a common factor of \(x\).
Factoring out this common \(x\), we simplify \(3x^2 + 2x\) to \((x(3x + 2))\). Factoring is a powerful tool in simplifying expressions and solving equations, making it easier to work with complex polynomials.

Combining Like Terms

Combining like terms is essential in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. For example, in the expression \((6x^2 + 2x - 3x^2)\), \(6x^2\) and \(-3x^2\) are like terms.
When combining like terms, simply add or subtract the coefficients while keeping the variable part unchanged: \(6x^2 - 3x^2 = 3x^2\).
This step often simplifies the expression significantly, reducing it to a more manageable form, as seen in our exercise, where \((6x^2 + 2x - 3x^2)\) simplifies to \((3x^2 + 2x)\).