Problem 80 - Intermediate Algebra

Question

Perform each division. \(\frac{8 x^{2}+12 x+9}{6}\)

Step-by-Step Solution

Answer

\(\frac{4x^2}{3} + 2x + \frac{3}{2}\)

Step 1: Identify the Division Operation

We are given an expression \( \frac{8x^2 + 12x + 9}{6} \). This expression implies that each term in the numerator, which consists of \(8x^2\), \(12x\), and \(9\), should be divided by 6 individually.

Step 2: Divide Each Term by 6

Divide the first term \(8x^2\) by 6: \( \frac{8x^2}{6} = \frac{4x^2}{3} \). For the second term \(12x\), divide it by 6: \( \frac{12x}{6} = 2x \). Finally, divide the constant term \(9\) by 6: \( \frac{9}{6} = \frac{3}{2} \).

Step 3: Write the Final Expression

Combine all the results from Step 2 to get the complete expression after division:\[ \frac{8x^2}{6} + \frac{12x}{6} + \frac{9}{6} = \frac{4x^2}{3} + 2x + \frac{3}{2} \].

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. In other words, they take the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These types of expressions are ubiquitous in algebra and calculus because they provide a way to represent division among polynomials.

• Understanding rational expressions is key to learning how to simplify and manipulate polynomial fractions.
• Common operations involving rational expressions include simplifying, multiplying, dividing, adding, and subtracting.

To perform operations on rational expressions, algebraic techniques such as factoring come in handy. Ensure that the denominator is never zero, and check for any restrictions on the variable that make the denominator zero. By practicing these steps, students will become proficient in managing these algebraic entities.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations like addition, multiplication, and division. Terms in an algebraic expression are separated by addition or subtraction signs, and each term is made up of numbers and variables raised to some power.
Here are some important points about algebraic expressions:

• They are essential in various branches of mathematics and are used to solve equations and model real-world problems.
• Algebraic operations include combining like terms and using the distributive property.

When solving problems involving algebraic expressions, it is crucial to simplify them where possible. Simplification often involves factoring, canceling common factors, and combining like terms. Being comfortable with algebraic expressions will allow students to tackle a wide variety of mathematical problems.

Simplifying Polynomials

Simplifying polynomials is the process of putting them into their simplest form. It involves several techniques, such as factoring, combining like terms, and performing polynomial division.
When simplifying polynomials, consider the following:

• Look for common factors in each term and simplify fractions where applicable.
• Factor polynomials completely. This involves the use of the distributive property and recognizing patterns like difference of squares or perfect square trinomials.

For the expression \( \frac{8x^2 + 12x + 9}{6} \), one must divide each term by the denominator. This process reduces the expression to \( \frac{4x^2}{3} + 2x + \frac{3}{2} \). This step-by-step approach ensures that the polynomial is accurately simplified while maintaining the integrity of the expression. Mastering these techniques brings students closer to understanding more complex algebraic manipulations.