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(3x^(2)-x-2)/(x^(2)-7x+12)- (3x^(2)-7x-6)/(x^(2)-4)

3 min read Jun 16, 2024
(3x^(2)-x-2)/(x^(2)-7x+12)- (3x^(2)-7x-6)/(x^(2)-4)

Simplifying the Expression: (3x^(2)-x-2)/(x^(2)-7x+12)- (3x^(2)-7x-6)/(x^(2)-4)

This expression involves rational expressions, which are fractions where the numerator and denominator are polynomials. To simplify it, we need to follow these steps:

1. Factor the denominators:

  • (x^(2)-7x+12) factors to (x-3)(x-4)
  • (x^(2)-4) factors to (x-2)(x+2)

2. Find the Least Common Multiple (LCM) of the denominators:

The LCM is the smallest expression that both denominators divide into. In this case, the LCM is (x-3)(x-4)(x-2)(x+2).

3. Rewrite each fraction with the LCM as the denominator:

  • (3x^(2)-x-2)/(x^(2)-7x+12) = (3x^(2)-x-2)(x-2)(x+2) / (x-3)(x-4)(x-2)(x+2)
  • (3x^(2)-7x-6)/(x^(2)-4) = (3x^(2)-7x-6)(x-3)(x-4) / (x-3)(x-4)(x-2)(x+2)

4. Simplify the numerators:

  • (3x^(2)-x-2)(x-2)(x+2) = (3x-2)(x+1)(x-2)(x+2)
  • (3x^(2)-7x-6)(x-3)(x-4) = (3x+2)(x-3)(x-4)

5. Combine the fractions:

((3x-2)(x+1)(x-2)(x+2) - (3x+2)(x-3)(x-4)) / (x-3)(x-4)(x-2)(x+2)

6. Expand and simplify the numerator:

After expanding the numerator and combining like terms, you'll get a polynomial expression.

7. Factor the simplified numerator:

  • If possible, factor the numerator to see if any common factors can be canceled with the denominator.

8. Final Simplified Expression:

After completing these steps, you will have the simplified form of the original expression.

Important Note: Remember that the original expression is undefined for values of x that make the denominator zero. In this case, x cannot be 3, 4, 2, or -2. These are the excluded values for the simplified expression.

This process may involve some complex algebraic manipulation, but by following the steps outlined above, you can systematically simplify the expression.

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