Problem 44 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{5 x^{5}}{x-2} \div \frac{10 x^{3}}{5 x-10} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5x^2}{2} \).

Step 1: Understanding the Division of Fractions

To simplify the expression \( \frac{5 x^{5}}{x-2} \div \frac{10 x^{3}}{5 x-10} \), we first interpret it as a division of fractions. Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. So, the expression becomes: \[ \frac{5 x^{5}}{x-2} \times \frac{5 x-10}{10 x^{3}} \].

Step 2: Simplifying the Second Fraction

Before proceeding with multiplication, simplify \( \frac{5 x-10}{10 x^{3}} \) if possible. The numerator \( 5x - 10 \) can be factored out as \( 5(x - 2) \). Hence, our fraction becomes \( \frac{5(x-2)}{10x^3} \).

Step 3: Setting Up the Multiplication

Now, substitute the simplified second fraction back into the multiplication setup: \[ \frac{5 x^{5}}{x-2} \times \frac{5(x-2)}{10x^3} \].

Step 4: Canceling Common Factors

In the multiplication of two fractions, we can cancel any common factors from the numerators and denominators. Notice that \( x-2 \) appears in both the numerator and the denominator, allowing us to cancel \( x-2 \). The expression simplifies to: \[ \frac{5x^5 \times 5}{10x^3} \].

Step 5: Further Simplification

Now review the simplified expression \( \frac{25x^5}{10x^3} \). Cancel the common factor of \( 5 \) in the numerator and denominator to get \( \frac{5x^5}{2x^3} \). Further simplify by dividing the powers of \( x \). We do this by subtracting the exponent in the denominator from the exponent in the numerator: \( x^{5-3} = x^2 \).

Step 6: Final Simplified Expression

This leads to the final simplified form of the expression: \( \frac{5x^2}{2} \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form while maintaining its equality. In algebra, this often includes tasks like reducing the complexity of expressions, bringing similar terms together, and eliminating any unnecessary components. Such simplifications help in understanding the nature of mathematical tasks and in solving algebraic equations more efficiently.

1. **Combining Like Terms**: In expressions, you often have some terms with the same variables or numbers. For example, in the expression \( 3x + 4x \), simplifying involves combining the terms to become \( 7x \).

2. **Reducing Fractions**: If an expression involves fractions, you can simplify it by canceling out common factors in the numerator and denominator. If you have \( \frac{6x}{9} \), divide numerator and denominator by \( 3 \) to get \( \frac{2x}{3} \).

3. **Using the Order of Operations**: Follow the established order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction), to ensure each step in simplifying is accurate and efficient.

Dividing Fractions

Dividing fractions is a key concept that can seem tricky at first, but it's much simpler once you understand the process. The essential idea is to transform a division into a multiplication problem, which is easier to solve.

1. **Reciprocal of a Fraction**: When you divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is simply flipping the numerator and the denominator. For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).

2. **Division as Multiplication**: Rather than division, you multiply by the reciprocal. If you have \( \frac{a}{b} \div \frac{c}{d} \), you rewrite it as \( \frac{a}{b} \times \frac{d}{c} \).

3. **Simplifying After Multiplication**: After rewriting the division as a multiplication, simplify by canceling any common factors between numerators and denominators before carrying out the multiplication.
By understanding this process, dividing fractions becomes a straightforward task, transforming into an opportunity to practice multiplication and simplification.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its simpler, more basic components or factors. This process is crucial for simplifying expressions and solving equations. It’s like unpacking a complex package to find more manageably-sized boxes inside.

1. **Identifying Common Factors**: Start by finding any common factors in the terms of the polynomial. For instance, in \( 2x^2 + 4x \), the common factor is \( 2x \), allowing us to factor it as \( 2x(x + 2) \).

2. **Using Special Formulas**: Some polynomials can be factored using special formulas, like the difference of squares \( a^2 - b^2 = (a - b)(a + b) \), or the perfect square trinomial \( a^2 + 2ab + b^2 = (a + b)^2 \). Recognizing these patterns speeds up the factoring process.

3. **Converting Larger Expressions**: When dealing with larger or more complex polynomials, it may be necessary to use techniques like grouping or the quadratic formula to factor them into their respective components.

Factoring transforms polynomials into a simpler form that is easier to work with, especially for integration within larger mathematical problems.