Answer:
the polynomial Ar³ + 3x² + bx - 3 can be rewritten as:
Ar³ + 3x² + (1/2)x - 3
Explanation:
Given that the polynomial has a factor of (2x + 3) and leaves a remainder of -3 when divided by (x + 2), we can use this information to find the values of a, b, and c.
1. Using the factor theorem:
The factor theorem states that if (2x + 3) is a factor of the polynomial, then the polynomial must equal zero when the factor is substituted into it. So we have:
(2x + 3) = 0
Solving for x, we get:
2x = -3
x = -3/2
2. Using the remainder theorem:
The remainder theorem states that if we divide the polynomial by (x + 2), the remainder we obtain is equal to the polynomial evaluated at the negation of the divisor. In this case, the remainder is given as -3. So we have:
P(-2) = -3
Substituting x = -2 into the polynomial, we can solve for a, b, and c:
(-2)^3 + 3(-2)^2 + b(-2) - 3 = -3
-8 + 12 - 2b - 3 = -3
1 - 2b = 0
-2b = -1
b = 1/2