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Question 1174083: Can someone please help me with this math riddle? I have been struggling with it for days now.
Can you help me think of a Polynomial (With using ax^2+bx+c) where a Polynomial of b, P(b) is divisible by a-b, and have the quotient of this be composite? I have tried so many different values for the a,b, and c in this polynomial, and I just don't know where to go off of here. Thank you so much for your time and help!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely, let's break down how to find such a polynomial.
**Understanding the Requirements**
* **Polynomial Form:** P(x) = ax² + bx + c
* **Divisibility:** P(b) must be divisible by (a - b).
* **Composite Quotient:** The result of P(b) / (a - b) must be a composite number (a number with factors other than 1 and itself).
**Steps to Find a Solution**
1. **Calculate P(b):**
* Substitute 'b' for 'x' in the polynomial: P(b) = ab² + b² + c
2. **Calculate the Divisor (a - b):**
* This is straightforward subtraction.
3. **Calculate the Quotient:**
* Divide P(b) by (a - b): (ab² + b² + c) / (a - b)
4. **Check for Composite Quotient:**
* The result of the division must be a composite number.
**Finding an Example**
Let's try some values:
* Let a = 6, b = 2, and c = 4.
* P(b) = (6 * 2²) + 2² + 4 = 24 + 4 + 4 = 32
* (a - b) = 6 - 2 = 4
* Quotient = 32 / 4 = 8
* 8 is a composite number (2 * 4).
**Therefore, the polynomial P(x) = 6x² + 2x + 4 satisfies the conditions.**
**Verification**
* P(2) = 6(2)² + 2(2) + 4 = 32
* a-b = 6-2 = 4
* 32/4 = 8.
* 8 is composite.
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