The expression x³ + 8x² + px - 25 leaves a remainder of R when divided by x - 1 and a remainder of -R when divided by x + 2.
Find the value of p
To find the value of p, we can use the Remainder Theorem. According to the theorem, when a polynomial f(x) is divided by x - a, the remainder is given by f(a).
Given that the expression x³ + 8x² + px - 25 leaves a remainder of R when divided by x - 1, we have:
f(1) = R
Substituting 1 into the expression, we get:
(1)³ + 8(1)² + p(1) - 25 = R
1 + 8 + p - 25 = R
p - 16 = R ----(1)
Similarly, when the expression is divided by x + 2 and the remainder is -R, we have:
f(-2) = -R
Substituting -2 into the expression, we get:
(-2)³ + 8(-2)² + p(-2) - 25 = -R
-8 + 32 - 2p - 25 = -R
-R - 1 = -2p
R + 1 = 2p ----(2)
Combining equations (1) and (2), we get:
p - 16 = R
R + 1 = 2p
Rearranging equation (2) to solve for R, we have:
R = 2p - 1
Substituting this value of R into equation (1), we get:
p - 16 = 2p - 1
p - 2p = 16 - 1
-p = 15
p = -15
Therefore, the value of p is -15.
To find the value of p, we can use the Remainder theorem. According to this theorem, if a polynomial f(x) is divided by x - a, then the remainder is equal to f(a).
We are given that the polynomial x³ + 8x² + px - 25 leaves a remainder of R when divided by x - 1. Using the remainder theorem, we can set x = 1 and find the remainder:
R = 1³ + 8(1)² + p(1) - 25
R = 1 + 8 + p - 25
R = p - 16
We are also given that the same polynomial leaves a remainder of -R when divided by x + 2. Again using the remainder theorem, we can set x = -2 and find the remainder:
-R = (-2)³ + 8(-2)² + p(-2) - 25
-R = -8 + 32 - 2p - 25
-R = -p - 1
Since we know that both remainders are equal, we can set the expressions for R equal to each other:
p - 16 = -p - 1
Adding p to both sides:
2p - 16 = -1
Adding 16 to both sides:
2p = 15
Dividing both sides by 2:
p = 7.5
Therefore, the value of p is 7.5.