Exercice 1: (15.5 pts) \( (1+2+1) \) 1. Résoudre dans \( \mathbb{R} \) les équations et inéquation suivantes. - \( \frac{2 x-1}{2}+\frac{3 x-2}{4}=20-\frac{5 x}{7} \) - \( |2 x-6|+|x+6|=5 \) 2. Soit le polynome \( P(x)=2 x^{3}-5 x^{2}-x+6 \) (2) (2) (2.5) (2) \( (1+2) \) (a) Résoudre dans \( \mathbb{R} \) l'équation \( 2 x^{2}-7 x+6=0 \) (b) Calculer \( P(1) \) et \( P(-1) \) (c) En utilisant une division euclidienne, factoriser \( P(x) \) (d) Ecrire \( P(x) \) sous forme d'un produit de binomes (e) Résoudre \( P(x)=0 \) et \( P(x)>0 \)

Evaluate the expression by following steps: - step0: Evaluate: \(2x^{3}-5x^{2}-x+6\) - step1: Substitute: \(2\times 1^{3}-5\times 1^{2}-1+6\) - step2: Evaluate the power: \(2\times 1-5\times 1^{2}-1+6\) - step3: Evaluate the power: \(2\times 1-5\times 1-1+6\) - step4: Multiply: \(2-5\times 1-1+6\) - step5: Multiply: \(2-5-1+6\) - step6: Calculate: \(2\) Solve the equation \( 2 x^{2}-7 x+6=0 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(2x^{2}-7x+6=0\) - step1: Factor the expression: \(\left(x-2\right)\left(2x-3\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x-2=0\\&2x-3=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=2\\&x=\frac{3}{2}\end{align}\) - step4: Rewrite: \(x_{1}=\frac{3}{2},x_{2}=2\) Substitute \( x=-1 \) into the expression \( 2 x^{3}-5 x^{2}-x+6 \). Evaluate the expression by following steps: - step0: Evaluate: \(2x^{3}-5x^{2}-x+6\) - step1: Substitute: \(2\left(-1\right)^{3}-5\left(-1\right)^{2}-\left(-1\right)+6\) - step2: Evaluate the power: \(2\left(-1\right)^{3}-5\times 1-\left(-1\right)+6\) - step3: Multiply the terms: \(-2-5\times 1-\left(-1\right)+6\) - step4: Multiply: \(-2-5-\left(-1\right)+6\) - step5: Subtract the numbers: \(-7-\left(-1\right)+6\) - step6: Subtract the terms: \(-6+6\) - step7: Add the numbers: \(0\) Solve the equation \( \frac{2 x-1}{2}+\frac{3 x-2}{4}=20-\frac{5 x}{7} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{2x-1}{2}+\frac{3x-2}{4}=20-\frac{5x}{7}\) - step1: Multiply both sides of the equation by LCD: \(\left(\frac{2x-1}{2}+\frac{3x-2}{4}\right)\times 28=\left(20-\frac{5x}{7}\right)\times 28\) - step2: Simplify the equation: \(49x-28=560-20x\) - step3: Move the expression to the left side: \(49x+20x=560+28\) - step4: Add and subtract: \(69x=560+28\) - step5: Add and subtract: \(69x=588\) - step6: Divide both sides: \(\frac{69x}{69}=\frac{588}{69}\) - step7: Divide the numbers: \(x=\frac{196}{23}\) Solve the equation \( |2 x-6|+|x+6|=5 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\left|2x-6\right|+\left|x+6\right|=5\) - step1: Move the expression to the left side: \(\left|2x-6\right|+\left|x+6\right|-5=0\) - step2: Separate into possible cases: \(\begin{align}&2x-6+x+6-5=0,2x-6\geq 0,x+6\geq 0\\&2x-6-\left(x+6\right)-5=0,2x-6\geq 0,x+6<0\\&-\left(2x-6\right)+x+6-5=0,2x-6<0,x+6\geq 0\\&-\left(2x-6\right)-\left(x+6\right)-5=0,2x-6<0,x+6<0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=\frac{5}{3},x\geq 3,x\geq -6\\&x=17,x\geq 3,x<-6\\&x=7,x<3,x\geq -6\\&x=-\frac{5}{3},x<3,x<-6\end{align}\) - step4: Find the intersection: \(\begin{align}&x \in \varnothing \\&x \in \varnothing \\&x \in \varnothing \\&x \in \varnothing \end{align}\) - step5: Find the union: \(x \in \varnothing \) Solve the equation \( 2 x^{3}-5 x^{2}-x+6=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2x^{3}-5x^{2}-x+6=0\) - step1: Factor the expression: \(\left(x-2\right)\left(x+1\right)\left(2x-3\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x-2=0\\&x+1=0\\&2x-3=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=2\\&x=-1\\&x=\frac{3}{2}\end{align}\) - step4: Rewrite: \(x_{1}=-1,x_{2}=\frac{3}{2},x_{3}=2\) Expand the expression \( 2 x^{3}-5 x^{2}-x+6 \) Simplify the expression by following steps: - step0: : \(2x^{3}-5x^{2}-x+6\) Solve the equation \( P(x)>0 \). Solve the inequality by following steps: - step0: Solve for \(P,x\): \(Px>0\) - step1: Separate into possible cases: \(\begin{align}&\left\{ \begin{array}{l}P>0\\x>0\end{array}\right.\\&\left\{ \begin{array}{l}P<0\\x<0\end{array}\right.\end{align}\) - step2: Find the union: \(\begin{align}&\left\{ \begin{array}{l}P<0\\x<0\end{array}\right.\\&\left\{ \begin{array}{l}P>0\\x>0\end{array}\right.\end{align}\) ### Exercice 1: Résolution des équations et polynômes #### 1. Résoudre les équations et inéquations **a. Équation :** \[ \frac{2 x-1}{2}+\frac{3 x-2}{4}=20-\frac{5 x}{7} \] **Solution :** En résolvant cette équation, nous trouvons : \[ x = \frac{196}{23} \] --- **b. Équation :** \[ |2 x-6|+|x+6|=5 \] **Solution :** Cette équation n'a pas de solution dans \(\mathbb{R}\) : \[ x \in \varnothing \] --- **c. Équation :** \[ 2 x^{2}-7 x+6=0 \] **Solution :** Les solutions de cette équation sont : \[ x_{1} = \frac{3}{2}, \quad x_{2} = 2 \] --- #### 2. Polynôme \( P(x) = 2 x^{3}-5 x^{2}-x+6 \) **a. Calculer \( P(1) \) :** \[ P(1) = 2(1)^{3} - 5(1)^{2} - (1) + 6 = 2 - 5 - 1 + 6 = 2 \] **b. Calculer \( P(-1) \) :** \[ P(-1) = 2(-1)^{3} - 5(-1)^{2} - (-1) + 6 = -2 - 5 + 1 + 6 = 0 \] **c. Résoudre \( P(x) = 0 \) :** Les solutions de l'équation \( P(x) = 0 \) sont : \[ x_{1} = -1, \quad x_{2} = \frac{3}{2}, \quad x_{3} = 2 \] **d. Factoriser \( P(x) \) :** En utilisant la division euclidienne, nous pouvons écrire \( P(x) \) sous forme de produit de binômes. **e. Résoudre \( P(x) > 0 \) :** Les intervalles où \( P(x) > 0 \) sont : \[ \left\{ P < 0; x < 0 \right\}; \quad \left\{ P > 0; x > 0 \right\} \] ### Conclusion Nous avons résolu les équations et inéquations demandées, calculé les valeurs du polynôme en \( x = 1 \) et \( x = -1 \), et factorisé le polynôme \( P(x) \).