(x-1)(x-2)(x2-9x+14)/(x-7)(x 2-3x+2)

Jasonbradley

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Jun 17, 2024

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Simplifying the Expression: (x-1)(x-2)(x^2-9x+14)/(x-7)(x^2-3x+2)

This expression involves a fraction with polynomials in both the numerator and denominator. To simplify it, we need to factor the polynomials and cancel out any common factors.

1. Factor the Polynomials

• Numerator:
  • (x-1)(x-2)(x^2-9x+14)
  • Notice that (x^2-9x+14) can be factored as (x-7)(x-2)
  • Therefore, the numerator becomes: (x-1)(x-2)(x-7)(x-2)
• Denominator:
  • (x-7)(x^2-3x+2)
  • (x^2-3x+2) can be factored as (x-2)(x-1)
  • The denominator becomes: (x-7)(x-2)(x-1)

2. Simplify the Expression

Now our expression is: [(x-1)(x-2)(x-7)(x-2)] / [(x-7)(x-2)(x-1)]

We can cancel out the common factors: (x-1), (x-2), and (x-7).

3. Final Result

After canceling out the common factors, we are left with (x-2).

Important Note: We have to be careful about the values of x that make the denominator zero, as this would make the expression undefined. In this case, x cannot be equal to 7, 2, or 1.

Therefore, the simplified form of the expression (x-1)(x-2)(x^2-9x+14)/(x-7)(x^2-3x+2) is (x-2), where x ≠ 7, 2, or 1.