To determine the relationship between the coefficients a, b, and c under the given condition, we begin by analyzing the divisibility criteria.
If the polynomial ax³ + bx² + c is divisible by x² + bx + c, it means that there exists some polynomial Q(x) such that:
ax³ + bx² + c = (x² + bx + c)Q(x)
The degree of the polynomial on the left (3) is equal to the degree of the polynomial on the right (2) plus the degree of Q(x) (which must be 1). This further implies that we can express Q(x) as:
Q(x) = kx + d for some constants k and d.
Expanding the right side gives:
(x² + bx + c)(kx + d) = kx³ + (b k + d)x² + (b d + c k)x + c d
Now, for divisibility to hold, the coefficients of corresponding powers of x must be equal. Setting up the equations:
- Coefficient of x³: a = k
- Coefficient of x²: b = b k + d
- Coefficient of x: 0 = b d + c k
- Constant term: c = c d
From the equation c = c d, we find that if c ≠ 0, then d = 1. If c = 0, we analyze the cases separately.
Next, substituting d = 1 into the second equation:
b = b k + 1
For this to hold true, rearranging gives:
b – b k = 1
b(1 – k) = 1
Now, from here, we can express k in terms of b:
k = 1 – rac{1}{b}
Now substituting k back into results, we get:
From the first equation, a = k = 1 – rac{1}{b}
Thus with this relation, we can compute ab:
ab = a imes b = igg(1 – rac{1}{b}igg)b = b – 1
This means that according to our findings, the product ab yields the result b – 1. Hence, while we have established a relationship between the two, the question implies we need further conditions to determine exact values of a and b. However, under the situation portrayed:
The relationship derived indicates that the product ab will result in b – 1.