Bunuel wrote:
Two prime numbers are considered consecutive if no other prime lies between them on the number line. If \(p_1\) and \(p_2\) are consecutive primes, with \(|p_1 – p_2| > 2\), what is the smallest possible absolute value of the coefficient of the x term in the distributed form of the expression \((x – p_1)(x – p_2)\)?
(A) 5
(B) 8
(C) 12
(D) 18
(E) 24
We're told that p1 and p2 are CONSECUTIVE primes (no primes in between)
We're also told that |p1 – p2| > 2, which means the two primes are MORE THAN 2 units away.
So, for example, the two primes cannot be 3 and 5, since the two primes are 2 units away.
Can two CONSECUTIVE primes be THREE units away?
For the primes to be THREE units away, one prime must be EVEN and the other must be ODD
Since 2 is the only EVEN prime, the smaller prime must be 2, which means the bigger prime must be 5.
HOWEVER, 2 and 5 are not CONSECUTIVE primes, since the prime number 3 lies between 2 and 5
So, we can conclude that two consecutive primes be CANNOT be THREE units away
Can two CONSECUTIVE primes be FOUR units away?
Let's check some primes that are FOUR units away
3 and 7. No good. They are not CONSECUTIVE since the prime number 5 lies between 3 and 7.
5 and 9. No good. 9 isn't prime.
7 and 11. PERFECT! 7 and 11 are prime, and they're consecutive.
So, 7 and 11 are the smallest values for p1 and p2.
What is the smallest possible absolute value of the coefficient of the x term in the distributed form of the expression (x – p1)(x – p2)?We can say that p1 = 7, and p2 = 11
We get: (x – p1)(x – p2) = (x – 7)(x – 11)
= x² - 11x - 7x + 77
= x²
- 18x + 77
The COEFFICIENT of the x term is
-18We want to find the smallest possible ABSOLUTE VALUE of the coefficient of the x term
So, we want |
-18|, which equals 18
Answer: D
Cheers,
Brent
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