Page 11 - ePD05201_升科大四技數學CII_課本PDF

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式的運算
1-1 多項式的四則運算

多項式恆等定理 2

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設 f x =(a 2)x +3(b +1) x +(2c 5)x + d，若 設 f x =3 a +2 x + b 3 x + 3c +5 x + d，若
f 1 = f 2 = f 3 = f 4 =5，則a 2b +3c 4d f 1 = f 2 = f 3 = f 4 = 5 ，則
= 。 2a b +3c d = 。
3 a +2 =0
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設 f x = ax + bx + cx + d，若 f 1 = f 2 b 3=0
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= f 3 = f 4 = k，則 f x = k a = b = c =0， ∵ f x =5 3c +5=0
d = k
d = 5
a = 2，b =3，3c = 5，d = 5
∵ f x 為至多三次多項式，
∴ 2a b +3c d = 7
但 f 1 = f 2 = f 3 = f 4 =5
a 2=0
3 b +1 =0
∴ f x =5
2c 5=0
d =5
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a =2，b = 1，c = ，d =5
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∴ a 2b +3c 4d =
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4 多項式的加減法

先將多項式依降（升）冪排列，再將同次項係數相加減。

註：採分離係數法時遇缺項要補

多項式的加減

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設 f x =12x +4x 23 + 6x 3 5x ， g x =2x 17 + 設 f x = 3 2x ， g x = x +2 ，則
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4x +22x +3x ，則 f x + g x =
f x + g x = f x g x = 。
f x g x = 。 ∵ f x = 8x +36x 2 54x +27，
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g x = x +8x +24x +32x +16
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將 f x 與 g x 依降冪排列再將同次項係數相 4 2
f x + g x = x +60x 22x +43
加減即可
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f x g x = x 4 16x +12x 2 86x +11
f x + g x =3x 6 5x +10x +34x +6x 40
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f x g x = 3x 6 5x +2x 3 10x +2x 6
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