(x-1)(x-2)(x^2-9x+14)/(x-7)(x^2-3x+2)

Jasonbradley

3 min read Jun 17, 2024

Simplifying Rational Expressions: A Step-by-Step Guide

This article will guide you through the process of simplifying the rational expression:

(x-1)(x-2)(x^2-9x+14) / (x-7)(x^2-3x+2)

1. Factoring the Expressions

First, we need to factor each of the quadratic expressions in the numerator and denominator.

Numerator:

(x^2 - 9x + 14): This factors into (x - 7)(x - 2)

Denominator:

(x^2 - 3x + 2): This factors into (x - 2)(x - 1)

Now, our expression looks like this:

(x-1)(x-2)(x-7)(x-2) / (x-7)(x-2)(x-1)

2. Identifying Common Factors

Next, we identify the common factors in the numerator and denominator:

(x-1)
(x-2)
(x-7)

3. Canceling Common Factors

We can cancel out the common factors, as long as they are not equal to zero. This means we need to exclude values of x that make any of the canceled factors equal to zero.

Our simplified expression is:

1 / 1 = 1

4. Restrictions

However, we need to consider the restrictions on the original expression. We cannot have values of x that would make the denominator equal to zero:

(x-7) = 0 => x = 7
(x-2) = 0 => x = 2
(x-1) = 0 => x = 1

Therefore, the simplified expression is 1, with the restrictions x ≠ 1, 2, 7.

Conclusion

By factoring and canceling common factors, we have successfully simplified the rational expression. Remember to consider any restrictions on the variable to ensure the expression is valid.