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

1 answer

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.