Puissances, racines, polynômes et fractions rationnelles#
Nom, prénom:
Note:
| Question | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Total |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Points | 2 | 2 | 6 | 2 | 5 | 2 | 7 | 2 | 2 | 2 | 32 |
| Obtenus |
Détails des calculs obligatoires. Attention au soin. Calculatrice non autorisée.
Question 1 (2 pts)#
Effectuez l'extraction de racine:
\(\sqrt{27} =\)
\(\sqrt{72} =\)
Solution
\(\sqrt{27} = \sqrt{3 \cdot 3 \cdot 3} = 3 \sqrt{3}\)
\(\sqrt{72} = \sqrt{2 \cdot 4 \cdot 9} = \sqrt{2} \cdot 2 \cdot 3 = 6\sqrt{2}\)
Question 2 (2 pts)#
Calculez et répondez en notation scientifique:
\(400\,000\,000 \cdot 0,000\,000\,6=\)
Solution
\(4 \cdot 10^8 \cdot 6 \cdot 10^{-7}=(4 \cdot 6) \cdot (10^8 \cdot 10^{-7})= 24 \cdot 10^{8-7}=24 \cdot 10^1=2.4 \cdot 10^1 \cdot 10^2=2.4 \cdot 10^2\)
Question 3 (6 pts)#
Effectuez les calculs suivants (réponses simplifiées et sans exposants négatifs):
\(\dfrac{x^{2} \cdot x^{5}}{x^{9}} =\)
\((-5)^4 \cdot (-2)^4 =\)
\((-3xy^3)^{3} =\)
\(\left (-\dfrac{7}{8} \right)^{-2} =\)
\(\sqrt{\dfrac{4}{15}} \cdot \sqrt{\dfrac{6}{10}}=\)
\(125^{\frac{1}{3}} =\)
Solution
\(\dfrac{x^{2} \cdot x^{5}}{x^{9}} = x^{2+5-9}=x^{-2}=\dfrac{1}{x^2}\)
\((-5)^4 \cdot (-2)^4 =(-10)^4= 10\,000\)
\((-3xy^3)^{3} =-27x^3y^9\)
\(\left (-\dfrac{7}{8} \right)^{-2} =\left (-\dfrac{8}{7} \right)^{2} =\dfrac{64}{49}\)
\(\sqrt{\dfrac{4}{15}} \cdot \sqrt{\dfrac{6}{10}}=\sqrt{\dfrac{4 \cdot 6}{15 \cdot 10}}=\dfrac{\sqrt{4}}{\sqrt{25}}=\dfrac{2}{5}\)
\(125^{\frac{1}{3}} =\sqrt[3]{125}=5\)
Question 4 (2 pts)#
Effectuez le calcul suivant (x est un nombre positif):
\((\sqrt{x} \cdot \sqrt[3]{x} \cdot \sqrt[4]{x})^2 =\)
Solution
\((\sqrt{x} \cdot \sqrt[3]{x} \cdot \sqrt[4]{x})^2 =(x^{\frac{1}{2}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{4}})^2 = (x^{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}})^2 = (x^{\frac{6 + 4+ 3}{12}})^2=(x^{\frac{13}{12}})^2=x^{\frac{13}{12} \cdot 2}=x^{\frac{13}{6}}\)
Question 5 (5 pts)#
Calculez et simplifiez:
\(-5x \cdot 4x^6=\)
\((6x^2 + 2x - 14x^2) - 3 + 6x - 8=\)
\(5x^2 - (2x - 5 - (2x^2 + 9x)) =\)
\((x + 3)^2 - 5x(2x - 1) =\)
Solution
\(-5x \cdot 4x^6= -20x^7\)
\((6x^2 + 2x - 14x^2) - 3 + 6x - 8=6x^2+2x-14x^2-3+6x-8=-8x^2+8x-11\)
\(5x^2 - (2x - 5 - (2x^2 + 9x))=5x^2-(2x - 5 - 2x^2 - 9x)=\) \(5x^2-2x+5+2x^2+9x=7x^2+7x+5\)
\((x + 3)^2 - 5x(2x - 1) =x^2+6x+9-10x^2+5x=-9x^2+11x+9\)
Question 6 (2 pts)#
Effectuez la division polynomiale avec reste:
\((x^3 - 5x + 7) : (x + 2) =\)
Solution
\((x^3 - 5x + 7) : (x + 2) =x^2-2x-1\) reste \(9\)
Question 7 (7 pts)#
Factorisez le plus possible:
\(x^2 + 14x + 49 =\)
\(x^2 - 64 =\)
\(x^2 - 8x + 12 =\)
\(3x^2 + 6x + 3 =\)
\(9x^5yz^2 - 18x^4y^3z^3 + 24x^2y^2z =\)
\(x^4 - 2x^2 + 1 =\)
\(xy + 2x + 3y + 6 =\)
Solution
\(x^2 + 14x + 49 =(x+7)^2\)
\(x^2 - 64 =(x+8)(x-8)\)
\(x^2 - 8x + 12 =(x-2)(x-6)\)
\(3x^2 + 6x + 3 =3(x^2+2x+1)=3(x+1)^2\)
\(9x^5yz^2 - 18x^4y^3z^3 + 24x^2y^2z =3x^2yz(3x^3z-6x^2y^2z^2+8y)\)
\(x^4 - 2x^2 + 1 =(x^2-1)^2=((x+1)(x-1))^2=(x+1)^2(x-1)^2\)
\(xy + 2x + 3y + 6 =x(y+2)+3(y+2)=(y+2)(x+3)\)
Question 8 (2 pts)#
Simplifiez la fraction suivante \(\dfrac{x^2 + 5x - 6}{3x + 18} =\)
Solution
\(\dfrac{x^2 + 5x - 6}{3x + 18} = \dfrac{(x+6)(x-1)}{3(x + 6)} =\dfrac{x-1}{3}\)
Question 9 (2 pts)#
Résolvez l'équation suivante par factorisation \(2x^3 = 32x\)
Solution
\(x = 0\) ou \(x = -4\) ou \(x = 4\)
\(S = \{-4; 0; 4\}\)
Question 10 (2 pts)#
Calculez et répondez sous forme d'une seule fraction simplifiée:
\(\dfrac{5x}{x + 3} - \dfrac{x + 2}{x^2 - 9} = \)
Solution
\(\dfrac{5x}{x + 3} - \dfrac{x + 2}{x^2 - 9} = \dfrac{5x}{x + 3} - \dfrac{x + 2}{(x+3)(x-3)}=\dfrac{5x(x-3)}{(x + 3)(x-3)} - \dfrac{x + 2}{(x+3)(x-3)}\) \(= \dfrac{5x(x-3)-(x+2)}{(x + 3)(x-3)} = \dfrac{5x^2-15x-x-2}{(x + 3)(x-3)} = \dfrac{5x^2-16x-2}{(x + 3)(x-3)}\)