Rose Little
09/11/2023 · Middle School
45. Soit le polynóme \( P \) défini sur \( \mathbb{R} \) par : \( P(x)=2 x^{3}+7 x^{2}+2 x-3 \). \( 1^{\circ} \) a) Calculer \( P(-1) \). b) Déduire de a) que \( P(x) \) peut \( s \) 'écrire pour tout nombre réel \( x \) sous la forme \( P(x)=(x+1) \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Substitute \( x=-1 \) into the expression \( 2x^3+7x^2+2x-3 \).
Evaluate the expression by following steps:
- step0: Evaluate:
\(2x^{3}+7x^{2}+2x-3\)
- step1: Substitute:
\(2\left(-1\right)^{3}+7\left(-1\right)^{2}+2\left(-1\right)-3\)
- step2: Evaluate the power:
\(2\left(-1\right)^{3}+7\times 1+2\left(-1\right)-3\)
- step3: Multiply the terms:
\(-2+7\times 1+2\left(-1\right)-3\)
- step4: Multiply:
\(-2+7+2\left(-1\right)-3\)
- step5: Multiply the numbers:
\(-2+7-2-3\)
- step6: Calculate:
\(0\)
a) En substituant \( x=-1 \) dans l'expression \( P(x)=2x^3+7x^2+2x-3 \), on obtient \( P(-1)=0 \).
b) En déduisant de \( P(-1)=0 \), on peut conclure que \( P(x) \) peut être écrit sous la forme \( P(x)=(x+1) \) pour tout nombre réel \( x \).
Quick Answer
a) \( P(-1)=0 \).
b) \( P(x)=(x+1) \) pour tout \( x \in \mathbb{R} \).
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