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Correction - Exercice 06 page 180 - Activités algébriques


1ère année secondaire

Activités algébriques

Exercice 06 page 180



Développons et réduisons les expressions suivantes
a) \((\sqrt{2}–\frac{1}{2})(2+\frac{\sqrt{2}}{2}+\frac{1}{4} )\).

\((\sqrt{2}–\frac{1}{2})(2+\frac{\sqrt{2}}{2}+\frac{1}{4} )\)

\(=2\sqrt{2}+\frac{\sqrt{2}\times\sqrt{2}}{2}+\frac{\sqrt{2}\times1}{4}-\frac{2}{2}-\frac{1\times\sqrt{2}}{2\times2}-\frac{1\times1}{2\times4}\)

\(=2\sqrt{2}+\frac{2}{2}+\frac{\sqrt{2}}{4}-\frac{2}{2}-\frac{\sqrt{2}}{4}-\frac{1}{8}\)

\(=2\sqrt{2}-\)\(\frac{1}{8}\)


b) \((\frac{1}{3}a–\frac{2}{5}b)(\frac{1}{3}a+\frac{2}{5}b)\).

\((\frac{1}{3}a–\frac{2}{5}b)(\frac{1}{3}a+\frac{2}{5}b)\)

\(=(\frac{1}{3}a)^2–(\frac{2}{5}b)^2\)

\(=\frac{1}{9}a^2–\frac{4}{25}b^2\)


c) \((2a–1)^2-(2a+1)(3a+5)\).

\((2a–1)^2-(2a+1)(3a+5)\)

\(=2.2a^2-2.2a.1+1^2-(2a\times3a+2a\times5+1\times3a+1\times5)\)

\(=4a^2-4a+1-(6a^2+10a+3a+5)\)

\(=4a^2-4a+1-(6a^2+13a+5)\)

\(=4a^2-4a+1-6a^2-13a-5)\)

\(=-2a^2-17a-4\)



d) \((2x–1)^3+3(2x–1)^2(1-x)+3(2x–1)(1–x)^2+(1–x)^3\).

\((2x–1)^3+3(2x–1)^2(1-x)+3(2x–1)(1–x)^2+(1–x)^3\)

\(=((2x-1)+(1-x))^3\)

\(=(2x-1+1-x)^3\)

\(=x^3\)



e) \((2-\frac{3}{4}t)^2+(\frac{1}{4}t+1)^2\).

\((2-\frac{3}{4}t)^2+(\frac{1}{4}t+1)^2\)

\(=2^2-2\times2\times\frac{3}{4}t+(\frac{3}{4}t)^2+(\frac{1}{4}t)^2+2\times\frac{1}{4}t\times1+1^2\)

\(=4-\frac{12}{4}t+\frac{9}{16}t^2+\frac{1}{16}t^2+\frac{2}{4}t+1\)

\(=\frac{10}{16}t^2-\frac{10}{4}t+5\)

\(=\frac{5}{8}t^2-\frac{5}{2}t+5\)
 


f) \((x–2)(x+3)(x^2-1)–(x^3+2x)(x–6)\).

\((x–2)(x+3)(x^2-1)–(x^3+2x)(x–6)\)

\(=(x^2+3x-2x-6)(x^2-1)-(x^4-6x^3+2x^2-12x)\)

\(=(x^2+x-6)(x^2-1)-x^4+6x^3-2x^2+12x)\)

\(=x^4-x^2+x^3-x-6x^2+6-x^4+6x^3-2x^2+12x\)

\(=7x^3-9x^2+11x+6\)



g) \((π+x)^3–3π(π +x)+2π^3–x^3\).

\((π+x)^3–3π(π +x)+2π^3–x^3\)

\(=\pi^3+3\pi^2x+3\pi x^2+x^3-3\pi^2-3\pi x+2\pi^3-x^3\)

\(=3\pi^3+3\pi^2x+3\pi x^2-3\pi^2-3\pi x\)







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