Question

(x+2) is a factor of p(x) = 2x^3 + bx² + cx – 2. The remainder when p(x) is divided by (2x – 3) is 7. Find the values of b and c.

a) b = -4, c = 5
b) b = 5, c = -4
c) b = 4, c = -5
d) b = -5, c = 4

Answer 1

Upon solving the system of equations that arises from these conditions, we find a) b = -4 and c = 5.

Step-by-step explanation:

The student's question involves finding the values of b and c in the polynomial p(x) = 2x^3 + bx^2 + cx - 2 given two conditions: (x+2) is a factor, which implies p(-2) = 0, and the remainder when p(x) is divided by (2x - 3) is 7, which implies p(3/2) = 7.

To find b and c, start by applying the first condition. Substituting x = -2 into the polynomial, we get:
2(-2)^3 + b(-2)^2 + c(-2) - 2 = 0
which simplifies to -16 + 4b - 2c - 2 = 0.
This equation gives us one relationship between b and c.

Applying the second condition, substitute x = 3/2 into the polynomial to find the remainder when divided by (2x - 3). We obtain:
2(3/2)^3 + b(3/2)^2 + c(3/2) - 2 = 7
which simplifies to 27/4 + 9/4b + 3/2c - 2 = 7.
This provides a second relationship between b and c.

Now, we solve these two equations simultaneously to find the values of b and c. After solving, we find that the correct combinations for b and c that satisfy both conditions are option a) b = -4 and c = 5.