急!急! F4 Polynomials 3

17.
From Mark's answer:
3x³ - 2x² - 7x - 2
= (x + 1)(3x² - 5x - 2)
= (x + 1)(x - 2)(3x + 1)

From Yvonne' answer :
3x³ - 2x² - 7x - 2
= (x - 2)(3x² + 4x + 1)
= (x - 2)(x + 1)(3x + 1)

f(x) should be factorized into (x +1)(x - 2)(3x + 1).
Mark and Yvonne gave differentanswers because both of them did NOT completely factorize f(x).


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21.
Let f(x) = x²(ax + 1) + bx + 3

The remainder when the f(x) is divided by 2x - 1 :
f(1/2) = a
(1/2)²[a(1/2) + 1] + b(1/2) + 3 = a
(1/8)a + (1/4) + (1/2)b + 3 = a
8[(1/8)a + (1/4) + (1/2)b + 3] = 8a
a + 2 + 4b + 24 = 8a
7a - 4b = 26 ...... [1]

The remainder when the f(x) is divided by x + 2 :
f(-2) = b
(-2)²[a(-2) + 1] + b(-2) + 3 = b
-8a + 4 - 2b + 3 = b
8a + 3b = 7 ...... [2]

[1]*3 + [2]*4 :
53a = 106
a = 2

Put a = 2 into [2] :
8(2) + 3b = 7
3b = -9
b = -3

Hence, a = 2 and b = -3

(b)
f(x) = x²(2x + 1) - 3x + 3
f(x) = 2x³ + x² - 3x + 3

The remainder when the polynomial is divided by x - a
= f(a)
= f(2)
= 2(2)³ + (2)² - 3(2) + 3
= 17