Question

Find the product. $$ \frac{x^3(x+5)}{x-9} \cdot \frac{(x-9)(x+8)}{3 x^3} $$

Short answer

The product is \(\frac{ (x^2 + 13x + 40)}{3}\)

Step-by-step solution

Rewrite the problem

To begin, rewrite the problem as follows: \( \frac{x^3 \cdot (x+5)}{x-9} \cdot \frac{(x-9) \cdot (x+8)}{3 \cdot x^3} \)

Cancel out

Now, look for common factors in the numerator and denominator: \( x^3 \) and \( x-9 \) can be divided from both top and bottom: \( \frac{(x+5)}{1} \cdot \frac{(x+8)}{3} \)

Simplify the expression

Multiply the two resulting numbers for the final answer: \((x+5) \cdot (x+8) \div 3\)

Distribute

In order to remove parentheses, distribute as follows: \(\frac{ (x \cdot x + x \cdot 8 + 5 \cdot x + 5 \cdot 8)}{3}\)

Simplify further

Simplify to get the final result: \(\frac{ (x^2 + 13x + 40)}{3}\).

Key concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying expressions that consist of coefficients and variables. In this exercise, we are dealing with products of rational expressions such as \( \frac{x^3(x+5)}{x-9} \) and \( \frac{(x-9)(x+8)}{3x^3} \). The basic idea is to multiply the numerators together and the denominators together. Here is how it works:

• First, multiply \( x^3 \) by \( (x+5) \) to expand the expression. This gives \( x^4 + 5x^3 \).
• For the second expression, multiply \( (x-9) \) by \( (x+8) \), which when expanded gives \( x^2 - x - 72 \).
• The next step involves multiplying the numerators together as a whole and likewise the denominators.

After performing these multiplications, always combine like terms for further simplification.

Factoring and Cancelation

Factoring and cancelation are crucial techniques for simplifying rational expressions, especially when dealing with polynomial multiplication. The goal is to reduce the expression to its simplest form by recognizing and dividing out common factors:

• In our exercise, both expressions have common factors \( x^3 \) and \( x-9 \) that appear in the numerator and denominator.
• Cancel these common factors. This effectively simplifies the expression, leading to fewer terms to multiply.
• Once these factors are removed, our new expression is \( \frac{(x+5)}{1} \cdot \frac{(x+8)}{3} \). This is much simpler and easier to manage.

Understanding the factoring process is essential to simplifying algebraic expressions effectively, allowing subsequent operations to be straightforward.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. The main goal is to perform operations like distribution and combination of like terms. Let's walk through the final steps:

• First, multiply \( (x+5) \cdot (x+8) \) resulting in \( x^2 + 8x + 5x + 40 \).
• Then, combine the like terms: \( x^2 + 13x + 40 \).
• Finally, divide the entire expression by 3, resulting in \( \frac{x^2 + 13x + 40}{3} \).

This thorough simplification ensures the expression is tidy and easy to interpret. Practicing these steps helps to identify and implement simplification strategies effectively in various algebraic problems.