Question

Perform the indicated operation and simplify the result. Leave your answer in factored form. $$ \frac{x^{2}+x-6}{x^{2}+4 x-5} \cdot \frac{x^{2}-25}{x^{2}+2 x-15} $$

Short answer

\( \frac{(x+3)(x-2)(x-5)}{(x-1)(x-3)} \)

Step-by-step solution

Factor each polynomial in the numerator and denominator

First, factor each polynomial to simplify the fractions: The numerator of the first fraction is: \( x^2 + x - 6 = (x + 3)(x - 2) \). The denominator of the first fraction is: \( x^2 + 4x - 5 = (x + 5)(x - 1) \). The numerator of the second fraction is: \( x^2 - 25 = (x + 5)(x - 5) \). The denominator of the second fraction is: \( x^2 + 2x - 15 = (x + 5)(x - 3) \).

Rewrite the expression with the factored forms

Now substitute the factored polynomials into the original expression: \[ \frac{(x+3)(x-2)}{(x+5)(x-1)} \times \frac{(x+5)(x-5)}{(x+5)(x-3)} \]

Cancel common factors

Identify and cancel the common factors in the numerator and denominator: Cancel \((x + 5)\) in the numerator and denominator. The resulting expression is: \[ \frac{(x+3)(x-2)}{(x-1)} \times \frac{(x-5)}{(x-3)} \]

Combine the remaining factors

Combine the remaining numerators and denominators: \[ \frac{(x+3)(x-2)(x-5)}{(x-1)(x-3)} \]

Key concepts

Factoring Polynomials

Factoring polynomials is like finding the pieces that multiply together to make the original polynomial. When you factor polynomials, you break them down into simpler expressions.
For example, to factor the polynomial numerator \( x^2 + x - 6 \), it is done by finding two numbers that multiply to give -6 and add up to 1.

• These numbers are +3 and -2, so \( x^2 + x - 6 = (x + 3)(x - 2) \).

This method applies to all kinds of polynomials.
For instance, another numerator, \( x^2 - 25 \), is a difference of squares, factored as \( (x + 5)(x - 5) \).

• Remember: look for special patterns, like the difference of squares \( a^2 - b^2 = (a+b)(a-b) \).
• When you have a trinomial, search for two numbers that multiply and add to the coefficients.

Multiplying Fractions

To multiply fractions, multiply the numerators (top parts) together and the denominators (bottom parts) together.
Using the factored forms, the expression \( \frac{(x+3)(x-2)}{(x+5)(x-1)} \times \frac{(x+5)(x-5)}{(x+5)(x-3)} \) becomes:

• Numerator: \( (x+3)(x-2) \times (x+5)(x-5) \)
• Denominator: \( (x+5)(x-1) \times (x+5)(x-3) \).

When you multiply, keep the terms separated; it makes canceling later easier.
Combining the remaining factors gives us \( \frac{(x+3)(x-2)(x-5)}{(x-1)(x-3)} \).

Canceling Common Factors

After factoring and before multiplying, look for common factors that appear in both the numerator and the denominator.
In the expression \( \frac{(x+3)(x-2)}{(x+5)(x-1)} \times \frac{(x+5)(x-5)}{(x+5)(x-3)} \),\( (x+5) \) is a common factor in the numerators and denominators.

• Cancel \( (x+5) \) to simplify the fractions.

After canceling, combine the remaining factors:

• Numerator: \( (x+3)(x-2)(x-5) \)
• Denominator: \( (x-1)(x-3) \)

Thus, the simplified expression is \( \frac{(x+3)(x-2)(x-5)}{(x-1)(x-3)} \).
Always factor completely and cancel out common terms for the simplest form.