Quiz 

Remainder & Factor Theorem

By remainder theorem,

    remainder = f(-2/3)
                   = 3(-2/3)3 - (-2/3)2 + 7(-2/3) + 5
                   = -1

Let x = 1

    2 - 1 - 2 + 3 = D
    D = 2
 
    Let x = -2
    -16 - 4 + 4 + 3 = -3C + 2
    3C = 15
    C = 5
 
    Let x = 0
    3 = B (-1)(2) -5 + 2
    -2B = 6
    B = -3
 
    Sub any value other than 1, 2 & 0 into x
    A = 2
 
 
2x3 - x2 - 2x + 3 ≡ (Ax + B)(x - 1)(x + 2) + C(x - 1) + D
Since x2 + x - 2 = (x - 1)(x + 2), (note the similarity?)
thus, remainder = C(x - 1) + D
                       = 5(x - 1) + 2
                       = 5x - 3

 

2x3 + x2 - 19x = 6

    let f(x) = 2x3 + x2 - 19x - 6
    (x - 3)        f(3) = 0
  (x - 3) is a factor of f(x)
 
    let f(x) = (x - 3)(Ax2 + Bx + C)
    A = 2
    C = 2
    let x = 1
    2 + 1 - 19 - 6 = -2(2 + B + 2)
    -22 = -8 - 2B
    B = 7
  f(x) = (x - 3)(2x2 + 7x + 2)
    2x2 + 7x + 2 = 0
    x = 
       = -3.19 or -0.31 (3 s.f.)
 
  x = -3.19 or -0.31 or 3 
 

let f(x) = x3 + px2 + qx + 6

    by remainder theorem, f(-1) = f(2)
    -1 + p - q + 6 = 8 + 4p + 2q + 6
    p - q - 5 = 4p + 2q + 14
    3p + 3q = 9 --------(1)
 
    by remainder theorem, f(-3) = -60
    -27 + 9p - 3q + 6 = -60
    3q = 39 + 9p --------(2)
    sub (2) into (1):
    3p + 39 + 9p = -9
    12p = -48
    p = -4
    q = 1