Question

When f(x) = (x - 1)³ + 6(px + 4)2 is divided by x + 2, the remainder is - 3. Find the values of p.​

Answer 1

The value for p will be equal to 3 / 4.

We have the function as:

f(x) = (x - 1)³ + 6(p x + 4) 2

f(x) = x³ - 3 x² + 3 x - 1 + 12 p x + 48

f(x) = x³ - 3 x² + 3 x ( 1 + 4 p) + 47

Now, it is divided by x + 2 and leaves a remainder -3.

This means that if we will subtract 3 from f(x), it will be perfectly divisible by x + 2.

g(x) = x³ - 3 x² + 3 x ( 1 + 4 p) + 47 - 3

g(x) = x³ - 3 x² + 3 x ( 1 + 4 p) + 44

Now put x = -2:

(-2)³ - 3 (-2)² + 3 (-2) ( 1 + 4 p) + 44 = 0

-8 - 12 - 6 ( 1 + 4 p) + 44 = 0

- 20 - 6 - 24 p + 44 = 0

24 p = 18

p = 3 / 4.

Therefore, the value for p will be equal to 3 / 4.

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