Answer:
29/7
Explanation:
If the polynomial P(x) has the same remainder when divided by x - 1 and by x + 3, then the difference between P(x) and its remainder when divided by x - 1 should be divisible by x - 1, and the difference between P(x) and its remainder when divided by x + 3 should be divisible by x + 3.
Let R1 and R2 be the remainders when P(x) is divided by x - 1 and by x + 3, respectively. Then, we have:
P(x) - R1 = (x - 1) Q(x) (1)
P(x) - R2 = (x + 3) S(x) (2)
where Q(x) and S(x) are polynomials.
Since P(x) is given as P(x) = 4x^3 - 3x^2 + bx + 6, we can substitute this expression into equations (1) and (2):
4x^3 - 3x^2 + bx + 6 - R1 = (x - 1) Q(x) (3)
4x^3 - 3x^2 + bx + 6 - R2 = (x + 3) S(x) (4)
To find the value of b that satisfies the problem condition, we need to set R1 = R2 and solve for b.
Subtracting equation (4) from equation (3), we get:
R1 - R2 = (x - 1) Q(x) - (x + 3) S(x)
Since R1 = P(1) and R2 = P(-3), we have:
P(1) - P(-3) = 4 - 3 + b(1 - 3) + 6 - (64 + 27 - 3b + 6)
Simplifying:
-57 + 4b = -28 - 3b
7b = 29
b = 29/7
Therefore, the value of b that will make P(x) have the same remainder when divided by x - 1 and by x + 3 is 29/7.