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If (x - 2) and (x + 1) are factors of the expression x3 + px2 + qx + 1, what is ...


Question

If (x - 2) and (x + 1) are factors of the expression x3 + px2 + qx + 1, what is the sum of p and q

Options

A) 9

B) -3

C) 3

D) \(\frac{17}{3}\)

E) \(\frac{2}{3}\)

The correct answer is B.

Explanation:

x3 + px2 + qx + 1 = (x - 1) Q(x) + R
x - 2 = 0, x = 2, R = 0,
4p + 2p = -9........(i)
x3 + px2 + qx + 1 = (x - 1)Q(x) + R
-1 + p - q + 1 = 0
p - q = 0.......(ii)
Solve the equation simultaneously
p = \(\frac{-3}{2}\)
q = \(\frac{-3}{2}\)
p + q = \(\frac{3}{2}\) - \(\frac{3}{2}\)
= \(\frac{-6}{2}\)
= -3

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Dicussion (1)

  • x3 + px2 + qx + 1 = (x - 1) Q(x) + R
    x - 2 = 0, x = 2, R = 0,
    4p + 2p = -9........(i)
    x3 + px2 + qx + 1 = (x - 1)Q(x) + R
    -1 + p - q + 1 = 0
    p - q = 0.......(ii)
    Solve the equation simultaneously
    p = \(\frac{-3}{2}\)
    q = \(\frac{-3}{2}\)
    p + q = \(\frac{3}{2}\) - \(\frac{3}{2}\)
    = \(\frac{-6}{2}\)
    = -3

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