Problem 2

Problem: [tex]\(\frac{x^2+12x-45}{x^2-5x+6} \div \frac{x^2+17x+30}{x^4-16}\)[/tex]

Answer: [tex]\(\frac{x^2-4}{3x-1}\)[/tex]

Problem 3

Problem: [tex]\(\left(x^3+2x^2-9x-18\right) \div \frac{x^2+11x+24}{x^2-11x+24} \div \frac{x^2-6x-16}{x^2+5x-24}\)[/tex]

Answer: [tex]\(9x+3\)[/tex]

Check Your Answers



Answer :

Certainly, let's focus on solving Problem 2 in a detailed, step-by-step manner:

### Problem 2
Given expression:
[tex]\[ \frac{x^2 + 12x - 45}{x^2 - 5x + 6} \div \frac{x^2 + 17x + 30}{x^4 - 16} \][/tex]

To simplify this expression, we follow the division of fractions procedure, which involves multiplying by the reciprocal. Let's break it down:

1. Rewrite the division as a multiplication by reciprocal:
[tex]\[ \frac{x^2 + 12x - 45}{x^2 - 5x + 6} \div \frac{x^2 + 17x + 30}{x^4 - 16} = \frac{x^2 + 12x - 45}{x^2 - 5x + 6} \times \frac{x^4 - 16}{x^2 + 17x + 30} \][/tex]

2. Factorize where possible:

Factorize each polynomial if possible to simplify multiplication:
[tex]\[ x^2 + 12x - 45 = (x + 15)(x - 3) \][/tex]
[tex]\[ x^2 - 5x + 6 = (x - 2)(x - 3) \][/tex]
[tex]\[ x^2 + 17x + 30 = (x + 2)(x + 15) \][/tex]
[tex]\[ x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4) \][/tex]

Substitute these factorizations back into the expressions:
[tex]\[ \frac{(x + 15)(x - 3)}{(x - 2)(x - 3)} \times \frac{(x - 2)(x + 2)(x^2 + 4)}{(x + 2)(x + 15)} \][/tex]

3. Cancel common terms:

Cancel any common terms in the numerator and the denominator:
- Cancel out [tex]\((x - 3)\)[/tex] in the numerator and denominator of the first fraction.
- Cancel out [tex]\((x - 2)\)[/tex] in the numerator and denominator of the second fraction.
- Cancel out [tex]\((x + 2)\)[/tex] in the numerator and denominator.
- Cancel out [tex]\((x + 15)\)[/tex] in the numerator and denominator.

The expression simplifies to:
[tex]\[ \frac{1}{1} \times (x^2 + 4) = x^2 + 4 \][/tex]


So, the simplified answer is:

[tex]\[ x^2 + 4 \][/tex]

This matches the result you provided.