For what value of b will the polynomial P(x) = 4x^3 - 3x^2 + bx + 6 have the same remainder when it is divided by x - 1 and by x + 3?
Hint:
P(x) will have the same remainder when divided by (x-1) and (x+3) if and only if
P(1)=P(-3).
To find the value of b for which the polynomial P(x) has the same remainder when divided by x - 1 and by x + 3, we can use the remainder theorem.
According to the remainder theorem, if a polynomial P(x) is divided by x - c, the remainder is equal to P(c). Therefore, in order for P(x) to have the same remainder when divided by both x - 1 and x + 3, we need to set the values of P(1) and P(-3) equal to find the value of b.
First, let's find the remainder of P(x) when divided by x - 1:
P(1) = 4(1)^3 - 3(1)^2 + b(1) + 6
= 4 - 3 + b + 6
= 7 + b
Now, let's find the remainder of P(x) when divided by x + 3:
P(-3) = 4(-3)^3 - 3(-3)^2 + b(-3) + 6
= -108 + 27 - 3b + 6
= -75 - 3b
Since we want the remainder to be the same, we can set P(1) = P(-3):
7 + b = -75 - 3b
Now, let's solve this equation for b:
b + 3b = -75 - 7
4b = -82
b = -82/4
b = -41/2
Therefore, the value of b for which the polynomial P(x) = 4x^3 - 3x^2 + bx + 6 has the same remainder when divided by x - 1 and by x + 3 is -41/2.
To find the value of b, we need to find the remainder when the polynomial P(x) is divided by both x - 1 and x + 3.
Let's start by finding the remainder when P(x) is divided by x - 1.
Using the remainder theorem, the remainder when P(x) is divided by x - 1 is equal to P(1). So, we substitute x = 1 into P(x):
P(1) = 4(1)^3 - 3(1)^2 + b(1) + 6
= 4 - 3 + b + 6
= 7 + b
Now, let's find the remainder when P(x) is divided by x + 3.
Using the remainder theorem again, the remainder when P(x) is divided by x + 3 is equal to P(-3). So, we substitute x = -3 into P(x):
P(-3) = 4(-3)^3 - 3(-3)^2 + b(-3) + 6
= -108 - 27b + 3b + 6
= -27b - 102
Since we want the remainder to be the same for both divisions, we set the two expressions equal to each other:
7 + b = -27b - 102
Simplifying the equation, we get:
28b = -109
Dividing both sides by 28, we find:
b = -109/28 ≈ -3.89
Therefore, the value of b that will make the remainder the same when P(x) is divided by x - 1 and by x + 3 is approximately -3.89.