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剰余の定理と因数定理の問題の解法 | 夢を叶える塾
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剰余の定理と因数定理の問題の解法

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剰余の定理と因数定理を使った問題です。

基本問題

(1)2x^2+3x-1x-3で割った時の余りを求めなさい。
(2)2x^2+3x-ax-3を因数に持つ。aの値を求めなさい。

解き方

(1)は剰余の定理を使いましょう。
整式P(x)x-aで割った時の余りはP(a)になります。

(2)は因数定理を使いましょう。
整式P(x)x-aを因数に持つとき、P(a)=0になります。

解説

(1)2x^2+3x-1x-3で割った時の余りを求めなさい。
剰余の定理を使います。

2x^2+3x-1x=3を代入します。
2(3)^2+3(3)-1=18+9-1=26
剰余の定理より、余りは26です。

(2)2x^2+3x-ax-3を因数に持つ。aの値を求めなさい。
因数定理を使います。

因数定理よりP(x)=2x^2+3x-ax-3を因数に持てばP(3)=0となります。
P(3)=2(3)^2+3(3)-a=18+9-a=0
aの方程式18+9-a=0を解きます。
a=27
答えです。

終わりに

因数定理は因数分解で大活躍します。
因数定理の前段である剰余の定理は少し忘れられがちです。
中々強力な定理なので使いこなせると良いですね。

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