Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$\left(3 x-\frac{1}{2}\right)(x+4)$$

Step-by-Step Solution

Answer

The solution in standard form is \(3x^2 + \frac{23}{2}x - 2\)

Step 1: Distribute Term by Term

To get started, distribute each term from the first parenthesis with every term in the second. \(3x * x + 3x * 4 - \frac{1}{2} * x - \frac{1}{2} * 4\)

Step 2: Simplify the Multiplication

Simplify the multiplication for each of the four terms you'll get: \(3x^2 + 12x - \frac{1}{2}x -2\) which can be written as \(3x^2 + \frac{23}{2}x - 2\).

Step 3: Write the Final Solution in Standard Form

The final answer in standard form is obtained by arranging the terms from highest power to lowest. Therefore, the answer is \(3x^2 + \frac{23}{2}x - 2\).

Key Concepts

Polynomial Operations

When working with polynomials, various operations can be performed, such as addition, subtraction, multiplication, and division. In the given exercise, we encounter multiplication of two binomials. Multiplication of polynomials involves using the distributive property to combine like terms.

For example, multiplying \(3x - \frac{1}{2}\) by \(x + 4\) requires distributing each term in the first polynomial to each term in the second, thus producing four separate products. The products are then simplified, and like terms are combined to form a polynomial in standard form with terms written in descending order of their degree.

Understanding this operation is crucial, as it lays the groundwork for more advanced algebraic manipulations and functions critical in calculus and other higher-level mathematics.

Distributive Property

The distributive property is a fundamental algebraic concept which states that \(a(b + c) = ab + ac\), meaning that a single term can be distributed to each term within a set of parentheses, and the sum of the products is equivalent to the original expression.

During the multiplication of polynomials, this property allows us to expand expressions in a systematic way, ensuring that each term from one polynomial is multiplied by each term of the other. It's pivotal in rearranging equations and simplifying expressions, having wide-reaching implications from arithmetic to algebra to advanced mathematics.

In the provided exercise, this property has been used to distribute \(3x\) across \(x + 4\) and \(\frac{1}{2}\) across the same, forming the basis for finding the resulting polynomial.

Simplifying Expressions

Simplifying expressions in algebra means reducing them to their least complex form without changing the value or the nature of the expression. After applying the distributive property, simplifying involves combining like terms—those with the same variable raised to the same power—and performing arithmetic on coefficients and constants.

In the context of the given exercise, after distributing terms, the expression includes like terms that can be simplified. This includes combining terms with \(x\) and constants separately. As a result, \(3x^2 + 12x - \frac{1}{2}x - 2\) simplifies to \(3x^2 + \frac{23}{2}x - 2\) by adding the coefficients of the terms involving \(x\). This simplification process is essential for readability, comparability, and when solving equations, as it brings us closer to the solution.