Question

Given that (x+2)(x+2) is a factor of p(x)=2x3+bx2+cx–2p(x)=2x3+bx2+cx–2 and the remainder when p(x)p(x) is divided by (2x–3)(2x–3) is 7, find the values of b and c.

a) b=5b=5, c=9c=9
b) b=9b=9, c=5c=5
c) b=7b=7, c=4c=4
d) b=4b=4, c=7c=7

Answer 1

To find the values of b and c, we use the fact that (x+2)^2 is a factor, implying p(-2) = 0, and that the remainder when dividing p(x) by (2x - 3) is 7, meaning p(3/2) = 7. Solving the resulting system of equations gives the values of b and c.

Step-by-step explanation:

The question involves finding the values of b and c in the cubic polynomial p(x) = 2x^3 + bx^2 + cx - 2 given that (x+2)^2 is a factor and the remainder when p(x) is divided by (2x - 3) is 7. We can use the factor theorem and polynomial long division or synthetic division to tackle this problem.

Since (x+2) is a factor, p(-2) = 0, which gives us one equation. Applying this, we get 2(-2)^3 + b(-2)^2 -2c - 2 = 0, which simplifies to -16 + 4b - 2c - 2 = 0.

Moreover, when divided by (2x - 3), the remainder is 7, so p(3/2) = 7. This gives us another equation: 2(3/2)^3 + b(3/2)^2 + c(3/2) - 2 = 7, simplifying to 27/4 + 9b/4 + 3c/2 - 2 = 7. Solving these two equations will yield the values for b and c.