F4 maths problem( Trigonometric )

用戶3618

2007-02-01 7:11 am

1. Show that x + 1 is the only linear factor of x^3 + 2x^2 + 4x + 3.

2. Solve 8cos^3θ + 8cos^2θ + 8cosθ +3 = 0 for 0° ＜ θ ＜ 360°
( Hint :Let x = 2cosθ )

回答 (1)

用戶80089

2007-02-01 7:22 am

✔ 最佳答案

1) Let f(x) = x^3 + 2x^2 + 4x + 3
f(-1) = -1 + 2 - 4 + 3 = 0
Thus, (x + 1) is a factor of f(x)
f(x) = (x + 1)(x^2 + x + 3)

discriminant of (x^2 + x + 3) = 1^2 - 4(1)(3) = -11 < 0
Thus, (x^2 + x + 3) cannot be factorized from quadratic factor to linear factor
Thus, (x + 1) is the only linear factor of f(x) = x^3 + 2x^2 + 4x + 3

2)8(cosθ)^3 + 8(cosθ)^2 + 8cosθ +3 = 0, Let x = 2cosθ
x^3 + 2x^2 + 4x +3 = 0
By Q1, x^3 + 2x^2 + 4x +3 = (x + 1)(x^2 + x + 3) = 0
x = -1 or no real solutions
2cosθ = -1
cosθ = -0.5
θ = 120°, 240°

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