When px^3 – 2px^2 + 3x – p is divided by (x – 1), the remainder is -5. Find p.
(2 marks)
To find the value of p, we can use the Remainder Theorem. According to the Remainder Theorem, when a polynomial f(x) is divided by (x – a), the remainder is equal to f(a).
In this case, we are given that the polynomial px^3 – 2px^2 + 3x – p is divided by (x – 1), and the remainder is -5. So, we can set up the equation:
px^3 – 2px^2 + 3x – p = (x – 1)(Q(x)) - 5
Here, Q(x) represents the quotient when the polynomial is divided by (x – 1).
Now, let's simplify the equation by expanding the right side:
px^3 – 2px^2 + 3x – p = Q(x)(x – 1) - 5
Next, we need to find the value of x that makes the equation true. We know that the remainder is -5, so we can substitute x = 1 into the equation:
p(1)^3 – 2p(1)^2 + 3(1) – p = Q(1)(1 – 1) - 5
This simplifies to:
p – 2p + 3 – p = 0 - 5
Combining like terms:
-2p + 3 = -5
Subtracting 3 from both sides:
-2p = -8
Dividing by -2:
p = 4
Therefore, the value of p is 4.