The polynomial can be factored as (x - 1)(x - 2)(x - 3), so x - 1 is one of its factors.

Substituting x = 1 into the polynomial gives f(1) = 1 - 4 + 6 - 4 + 1 = 0.

The polynomial can be rewritten in a form that shows symmetry about the line x = 1.

A polynomial has a double root when its discriminant is equal to zero.

Using Vieta's formulas, the sum of the roots (-2 + 3) = 1, which means b = -1.

Using Vieta's formulas, the sum of the roots (3 + (-2)) = 1, hence b = -1.

According to Vieta's formulas, the sum of the roots r1 + r2 of the polynomial x^2 - 5x + 6 is equal to the coefficient of x (which is -(-5)) = 5.

Factoring the polynomial gives (x - 2)(x - 3), so the roots are 2 and 3.

Factoring the polynomial P(x) gives (x - 2)(x - 3), so the roots are 2 and 3.

Calculating P(1) gives 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.

Let the original price be x. The new price after a 10% increase is x + 0.1x = 1.1x. Setting this equal to $22 gives us 1.1x = 22, so x = 22/1.1 = 20. Therefore, the original price was $20.

Let the original price be $100. After a 10% increase, the price becomes $110. After a 10% decrease, the price becomes $110 - $11 = $99. The net change is -1%, so the answer is 0%.

Let the original price be x. Then, x + 0.2x = 24, or 1.2x = 24. Thus, x = 24 / 1.2 = 20.

Let the original price be Rs. 100. After a 20% increase, price = 120. After a 20% decrease, price = 120 - 24 = 96. Net change = (96 - 100)/100 * 100 = -4%.

Let the original price be $100. After a 20% increase, the price becomes $120. After a 20% decrease, the price becomes $120 - 0.20 × 120 = $96. The net change is (96 - 100)/100 × 100 = -4%.

Let the original price be 100. After a 20% increase, the price is 120. After a 20% decrease, the price is 96. Net change = (96-100)/100 * 100% = -4%.

Let the original price be 100. After a 20% increase, price = 120. After a 20% decrease, price = 120 - 24 = 96. Net change = (96 - 100)/100 * 100% = -4%.

The interest earned is $400. Using SI = PRT, we have 400 = 2000 * R * 3. Solving for R gives R = 6.67%.

Using A = P + SI, where SI = P * r * t / 100, we have SI = 2000 * 4 * 5 / 100 = $400. Thus, A = 2000 + 400 = $2400.

Using the simple interest formula, Total Amount = Principal + SI = 800 + (800 * 4 * 3) / 100 = $840.

Total amount = Principal + SI = 800 + (800 * 6/100 * 4) = $1040.

Simple Interest = P * r * t = 5000 * 0.12 * 3 = 1800. Total Amount = Principal + Interest = 5000 + 1800 = 6800.

The probability of failing is 1 - P(passing) = 1 - 0.75 = 0.25.

The probability of either A or B occurring is P(A) + P(B) - P(A and B) = 0.2 + 0.5 - (0.2 * 0.5) = 0.7.

For independent events, P(A and B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.

For mutually exclusive events, P(C or D) = P(C) + P(D) = 0.2 + 0.3 = 0.5.

The probability that it will not rain is 1 - 0.7 = 0.3.

The product of 6 and 7 is 42, while 8 and 9 gives 72. Therefore, the correct pair is 8 and 9.

6 and 7 are consecutive integers whose product is 42, not 72.

The other number can be found by dividing the product by the known number: 144 / 12 = 12.