Problem 79 - Elementary Algebra

Question

Problem: Simplify: \(\left(24 x^{3}+8 x^{2}-2 x\right) \div(4 x)\) Incorrect Answer: \(\frac{24 x^{3}+8 x^{2}-2 x}{4 x}\) $$ \begin{aligned} &=\frac{6 \cdot 4 \cdot x \cdot x^{2}+2 \cdot 4 x^{2}-2 x}{4 \cdot x} \\ &=6 x^{2}+8 x^{2}-2 x \\ &=14 x^{2}-2 x \end{aligned} $$

Step-by-Step Solution

Answer

The simplified expression is \( 6 x^2 + 2 x - \frac{1}{2} \).

Step 1: Write the given expression

The problem is to simplify the expression \( \left(24 x^3 + 8 x^2 - 2 x \right) \div (4 x) \). First, rewrite the division as a fraction: \( \frac{24 x^3 + 8 x^2 - 2 x}{4 x} \).

Step 2: Distribute the denominator

Separate each term in the numerator by the denominator: \( \frac{24 x^3}{4 x} + \frac{8 x^2}{4 x} - \frac{2 x}{4 x} \).

Step 3: Simplify each fraction

Simplify the fractions step-by-step: For \( \frac{24 x^3}{4 x} \), divide 24 by 4 and subtract the exponents for x: \( 6 x^2 \). For \( \frac{8 x^2}{4 x} \), divide 8 by 4 and subtract the exponents for x: \( 2 x \). For \( \frac{2 x}{4 x} \), divide 2 by 4 resulting in \( \frac{1}{2} \) and cancel out x: \( \frac{1}{2} \).

Step 4: Write the simplified expression

Combine the simplified terms: \( 6 x^2 + 2 x - \frac{1}{2} \).

Key Concepts

Fractional Division

Fractional division in math is a method used to divide one fraction by another. You can also use it when dividing algebraic expressions by interpreting them as fractions. In our problem, we start with the expression: \( \left(24 x^3 + 8 x^2 - 2 x \right) \div (4 x)\).

This division can be rewritten as a fraction: \( \frac{24 x^3 + 8 x^2 - 2 x}{4 x} \). To simplify this, we need to divide each term in the numerator by the denominator individually. This means:

• \( \frac{24 x^3}{4 x} \)
• \( \frac{8 x^2}{4 x} \)
• \( \frac{2 x}{4 x} \)

We will handle these divisions one by one.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial expression to its simplest form. This is done by performing operations like division while adhering to the properties of exponents and coefficients.

Let's break down the simplification of each fraction from our example:

• For \( \frac{24 x^3}{4 x} \), divide 24 by 4 to get 6, and then apply the rule for dividing exponents: \( x^{3-1} = x^2 \). So, it becomes \( 6x^2 \).
• For \( \frac{8 x^2}{4 x} \), divide 8 by 4 to get 2, and then use the exponent rule \( x^{2-1} = x \). This simplifies to \( 2x \).
• For \( \frac{2 x}{4 x} \), we divide 2 by 4 to get \( \frac{1}{2} \) and cancel out the \( x \) terms. We are left with \( \frac{1}{2} \).

This gives us all the simplified terms that we combine in the last step: \( 6x^2 + 2x - \frac{1}{2} \).

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. It states that \( a(b + c) = ab + ac \). In the context of our problem, we use the distributive property in the reverse form to distribute the denominator across each term in the numerator.

With our expression, this looks like:

\( \frac{24 x^3 + 8 x^2 - 2 x}{4 x} = \frac{24 x^3}{4 x} + \frac{8 x^2}{4 x} - \frac{2 x}{4 x} \).

Each term in the numerator is divided individually by \( 4x \), as previously illustrated.
This method can simplify even more complex expressions by breaking them into manageable parts.