Problem 72 - Intermediate Algebra

Question

Perform the operations and simplify. $$ \frac{x^{3}+1}{4} \div \frac{x+1}{2} $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite Division as Multiplication

To simplify the expression, start by rewriting the division problem as a multiplication problem. Use the property \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \). Thus, \( \frac{x^3 + 1}{4} \div \frac{x+1}{2} \) becomes \( \frac{x^3 + 1}{4} \times \frac{2}{x+1} \).

Step 2: Combine the Fractions

Now that we have a multiplication problem, multiply the numerators together and the denominators together. This gives us: \[ \frac{(x^3 + 1) \times 2 }{4 \times (x+1)} = \frac{2(x^3 + 1)}{4(x+1)}. \]

Step 3: Simplify the Expression

Simplify the expression \( \frac{2(x^3 + 1)}{4(x+1)} \). Notice that \( x^3 + 1 \) can be factored using the sum of cubes formula: \( x^3 + 1 = (x+1)(x^2-x+1) \). Substituting this back into the expression, we have: \[ \frac{2((x+1)(x^2-x+1))}{4(x+1)}. \]

Step 4: Cancel Common Factors

In the fraction \( \frac{2((x+1)(x^2-x+1))}{4(x+1)} \), \( (x+1) \) is a common factor in the numerator and the denominator. Cancel \( (x+1) \) to simplify the expression: \[ \frac{2(x^2-x+1)}{4}. \]

Step 5: Reduce the Coefficients

The coefficients \(2\) and \(4\) can be simplified: \( \frac{2}{4} = \frac{1}{2} \). Therefore, the expression simplifies to: \( \frac{1}{2}(x^2-x+1) \). This can be left in its factored form or expanded: \[ \frac{x^2}{2} - \frac{x}{2} + \frac{1}{2}. \]

Key Concepts

Fraction Division

Understanding fraction division is key to many algebra problems. When dividing fractions, remember to multiply by the reciprocal of the divisor. For example, if we have \( \frac{a}{b} \div \frac{c}{d} \), it is the same as \( \frac{a}{b} \times \frac{d}{c} \). This works because dividing by a fraction is equivalent to multiplying by its inverse.

In our exercise, the division \( \frac{x^3 + 1}{4} \div \frac{x + 1}{2} \) converts into the multiplication \( \frac{x^3 + 1}{4} \times \frac{2}{x + 1} \). This step changes a division problem into a multiplication one, which is often simpler to handle with fractions.

Multiplication of Fractions

Once you've rewritten a division problem as multiplication, combining the fractions becomes straightforward. Fractions are multiplied by multiplying their numerators together and their denominators together. In the exercise, after converting the division into multiplication, we obtain: \[ \frac{(x^3 + 1) \times 2}{4 \times (x + 1)}. \]

Here's how to do it:

• Multiply the numerators: \( (x^3 + 1) \times 2 = 2(x^3 + 1) \).
• Multiply the denominators: \( 4 \times (x + 1) = 4(x + 1) \).

This method keeps operations organized and sets the stage for simplifying the resulting expression.

Simplifying Algebraic Expressions

Simplifying is about making an expression more manageable by eliminating unnecessary parts. In our exercise, we start with \( \frac{2(x^3 + 1)}{4(x + 1)} \).

To simplify:

• First, recognize any patterns - here, \( x^3 + 1 \) is a sum of cubes, which factors into \( (x+1)(x^2-x+1) \).
• Substitute this back into the fraction, giving \( \frac{2((x+1)(x^2-x+1))}{4(x+1)} \).

Finding these factors is crucial because it reveals common factors that can be canceled in order to tidy up the expression even more.

Factoring Algebraic Expressions

Factoring is an essential skill for simplifying and solving algebraic expressions. In the context of simplifying fractions, it helps in identifying terms that can be canceled.

Consider the expression \( x^3 + 1 \), which uses the sum of cubes formula: \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \). With \( a = x \) and \( b = 1 \), it becomes \( (x+1)(x^2-x+1) \).

This allows for the expression to be rewritten and simplified:

• Cancel out \( (x+1) \) in the numerator and denominator: \( \frac{2((x+1)(x^2-x+1))}{4(x+1)} \Rightarrow \frac{2(x^2-x+1)}{4} \).
• Always check for commonality in expressions—eliminating them refines the solution.

Finally, reduce any coefficients: \( \frac{2}{4} = \frac{1}{2} \), leaving us with a clean and simple form.