Let f(x) =ax^3+bx^2-25x+12, If f(x)is divisible by x^2+x-12
a.find two linear factors of f(x)
b.find the values of a and b
c.use the resuls of (a) and (b) to factorize f(x)
數學問題 40points
2009-12-27 6:51 pm
回答 (3)
2009-12-27 7:14 pm
✔ 最佳答案
(a) The factor is x^2 + x - 12= (x - 3)(x + 4)
The 2 linear factors are x - 3 and x + 4
(b) By Remainder Theorem,
f(3) = 0 => 27a + 9b - 75 + 12 = 0
27a + 9b - 63 = 0
3a + b - 7 = 0 ... (1)
f(-4) = 0 => -64a + 16b + 100 + 12 = 0
-64a + 16b + 112 = 0
-4a + b + 7 = 0 ... (2)
(1) - (2) => 7a - 14 = 0 => a = 2
Sub into (2), -8 + b + 7 = 0 => b = 1
(c) f(x) = 2x^3 + x^2 - 25x + 12
By long division, f(x) / (x^2 + x - 12) = (2x - 1)
Therefore f(x) = (2x - 1)(x - 3)(x + 4)
2009-12-29 3:34 am
................-1+2x
...............________________
-12+x+x^2)12-25x+bx^2+ax^3
...............)12 -x - x^2
...............
...............) -24x +(b+1)x^2+ax^3
................) -24x + 2 x^2+2x^3
.................-------
...................0
So a=2,b=1
...............________________
-12+x+x^2)12-25x+bx^2+ax^3
...............)12 -x - x^2
...............
...............) -24x +(b+1)x^2+ax^3
................) -24x + 2 x^2+2x^3
.................-------
...................0
So a=2,b=1
2009-12-28 12:02 am
a) The factor is x^2 + x - 12
= (x - 3)(x + 4)
The 2 linear factors are x - 3 and x + 4
(b) By Remainder Theorem,
f(3) = 0 => 27a + 9b - 75 + 12 = 0
27a + 9b - 63 = 0
3a + b - 7 = 0 ... (1)
f(-4) = 0 => -64a + 16b + 100 + 12 = 0
-64a + 16b + 112 = 0
-4a + b + 7 = 0 ... (2)
(1) - (2) => 7a - 14 = 0 => a = 2
Sub into (2), -8 + b + 7 = 0 => b = 1
(c) f(x) = 2x^3 + x^2 - 25x + 12
By long division, f(x) / (x^2 + x - 12) = (2x - 1)
Therefore f(x) = (2x - 1)(x - 3)(x + 4)
= (x - 3)(x + 4)
The 2 linear factors are x - 3 and x + 4
(b) By Remainder Theorem,
f(3) = 0 => 27a + 9b - 75 + 12 = 0
27a + 9b - 63 = 0
3a + b - 7 = 0 ... (1)
f(-4) = 0 => -64a + 16b + 100 + 12 = 0
-64a + 16b + 112 = 0
-4a + b + 7 = 0 ... (2)
(1) - (2) => 7a - 14 = 0 => a = 2
Sub into (2), -8 + b + 7 = 0 => b = 1
(c) f(x) = 2x^3 + x^2 - 25x + 12
By long division, f(x) / (x^2 + x - 12) = (2x - 1)
Therefore f(x) = (2x - 1)(x - 3)(x + 4)
參考: me
收錄日期: 2021-04-23 23:21:23
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