Problem 82 - Elementary Algebra

Question

Problem: Simplify: \(\left(6 x^{2}+8 x-3\right) \div(3 x)\) Incorrect Answer: \(\left(6 x^{2}+8 x-3\right) \div(3 x)\) $$ \begin{aligned} &=\frac{6 x^{2}}{3 x}+\frac{8 x}{3 x}-\frac{3}{3 x} \\ &=2 x+\frac{8}{3}-x \\ &=x+\frac{8}{3} \end{aligned} $$

Step-by-Step Solution

Answer

The simplified form is \(2x + \frac{8}{3} - \frac{1}{x}\).

Step 1: Identify Terms in the Numerator

Recognize that the numerator is the polynomial \(6x^2 + 8x - 3\). It consists of three separate terms: \(6x^2\), \(8x\), and \(-3\).

Step 2: Separate the Division for Each Term

Divide each term in the polynomial by the denominator \(3x\). This gives us three separate fractions: \[ \frac{6x^2}{3x}, \frac{8x}{3x}, \frac{-3}{3x} \].

Step 3: Simplify Each Fraction

Simplify each fraction: \[ \frac{6x^2}{3x} = 2x, \frac{8x}{3x} = \frac{8}{3}, \frac{-3}{3x} = \frac{-1}{x} \]. Note: \( \frac{-3}{3x} = \frac{-1}{x} \) instead of the incorrect \( \frac{-3}{3x} \).

Step 4: Combine Simplified Terms

Combine the simplified terms together: \(2x + \frac{8}{3} - \frac{1}{x}\).

Key Concepts

Simplifying Polynomials

Polynomials are expressions made up of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents of variables. Simplifying polynomials means reducing them to their simplest form. This involves combining like terms and performing operations within the polynomial. For example, in the problem \(\left(6 x^{2}+8 x-3\right) \div(3 x)\), we reduce the polynomial by dividing each term by the given denominator.
Start by breaking down each term in the numerator and simplifying them individually. This helps make the complex expression simpler and easier to handle.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Just like numerical fractions, these can be simplified by dividing the numerator and the denominator by their greatest common factor (GCF).
In the given problem, \(6x^2 + 8x - 3 \div 3x\), you handle each term in the polynomial separately, simplifying over a common term. By dividing each term of the polynomial by \(3x\), you simplify the fraction and make the expression clearer and more manageable.

• Identify common factors in the numerator and the denominator.
• Divide each term of the polynomial in the numerator by the common denominator.
• Combine the resulting terms to get the simplified expression.

This process ensures the expression is reduced to its simplest form.

Algebraic Fractions

Algebraic fractions are fractions that contain algebraic expressions in the numerator, the denominator, or both. Simplifying algebraic fractions is crucial for solving algebraic equations and performing algebraic manipulations.
The steps are similar to simplifying numeric fractions but involve more algebraic manipulation. In our particular example: \(\frac{6 x^{2}}{3 x}, \frac{8 x}{3 x}, \frac{-3}{3 x}\), each term is simplified individually: \(\frac{6x^2}{3x} = 2x\), \(\frac{8x}{3x} = \frac{ 8}{3}\), and \(\frac{-3}{ 3x} = \frac{-1}{x}\).
This simplifies the initial complex algebraic fraction into simpler components that are easier to work with and understand.

Elementary Algebra

Elementary Algebra is the study of basic algebraic concepts, including operations on variables, solving equations, and simplifying expressions. It lays the foundation for understanding more complex algebraic problems.
In the problem \(6x^2 + 8x - 3 \div 3x\), elementary algebra principles are applied to simplify the expression by dividing each term individually by the divisor.

• Recognize and separate each term in the polynomial.
• Perform individual division on each term.
• Combine the simplified results.

This showcases fundamental algebraic skills that help in tackling larger, more complex algebra problems effectively.