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.
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
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