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- \documentclass[11pt]{article}
- \usepackage[utf8]{inputenc}
- \usepackage[T1]{fontenc}
- \usepackage[francais]{babel}
- \usepackage[top=1.5cm, bottom=1.5cm, left=1.5cm, right=1.5cm]{geometry}
- \usepackage{amsmath,amssymb,mathrsfs}
- \pagestyle{empty}
- \begin{document}
- Simon \bsc{Léonard} \FrenchEnumerate{2}8
- \fontfamily{phv}\selectfont
- \begin{center}\LARGE{Devoir maison}\end{center}
- {\Large Exercice 25p141}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- M &= (23x+1) (-17x+1) + (23x+1)^2\\
- &= (23x+1) \bigl[(-17x+1) + (23x+1)\bigr]\\
- &= (23x+1) (6x+2)
- \end{align*}
- \begin{align*}
- A &= (13x-14)(25x-11)-(13x-14)^2\\
- &= (13x-14) \bigl[(25x-11)-(13x-14)\bigr]\\
- &= (13x-14) (25x-11-13x+14)\\
- &= (13x-14) (12x+3)
- \end{align*}
- \begin{align*}
- N &= (8-18x)^2-(16x-3)(8-18x)\\
- &= (8-18x) \bigl[(8-18x)-(16x-3)\bigr]\\
- &= (8-18x) (8-18x-16x+3)\\
- &= (8-18x) (-34x+11)
- \end{align*}
- \begin{align*}
- U &= (11-2x)(9-7x)+(2x-11)(11x+2)\\
- &=(11-2x)(9-7x)-(11-2x)(11x+2)\\
- &= (11-2x)\bigl[(9-7x)-(11x+2)\bigr]\\
- &= (11-2x)(9-7x-11x-2)\\
- &= (11-2x)(-18x+7)
- \end{align*}
- \end{minipage}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- E &= (6x+23)(6x-5)-(19x-6)(5-6x)\\
- &=(6x+23)(6x-5)+(6x-5)(19x-6)\\
- &=(6x-5)(6x+23+19x-6)\\
- &= (6x-5)(25x+17)
- \end{align*}
- \begin{align*}
- L &= (16x+13)(21x-3)+(16x+13)\\
- &= (16x+13)(21x-3+1)\\
- &= (16x+13)(21x-2)
- \end{align*}
- \begin{align*}
- S&=(-14x+5)-(4x-7)(-14x+5)\\
- &=(-14x+5)(1-4x+7)\\
- &= (-14x+5)(-4x+8)
- \end{align*}
- \end{minipage}
- {\Large Exercice 26p141}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- H&=x^2-121\\
- &= (\sqrt{x^2}+\sqrt{121})(\sqrt{x^2}-\sqrt{121})\\
- &= (x+11)(x-11)
- \end{align*}
- \begin{align*}
- E&=(x-4)^2-36\\
- &=\Bigl[\sqrt{(x-4)^2}+\sqrt{36}\Bigr]\\
- &\quad\times\Bigl[\sqrt{(x-4)^2}-\sqrt{36}\Bigr]\\
- &=(x-4+6)(x-4-6)\\
- &=(x+2)(x-10)
- \end{align*}
- \begin{align*}
- R&=x^2-5\\
- &= (\sqrt{x^2}+\sqrt{5})(\sqrt{x^2}-\sqrt{5})\\
- &= (x+\sqrt{5})(x-\sqrt{5})
- \end{align*}
- \end{minipage}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- T&=25-(2-x)^2\\
- &=\Bigl[\sqrt{25}+\sqrt{(2-x)^2}\Bigr]\Bigl[\sqrt{25}-\sqrt{(2-x)^2}\Bigr]\\
- &=(5+2-x)(5-2+x)\\
- &=(7-x)(3+x)
- \end{align*}
- \begin{align*}
- Z&=(x+3)^2-(2x+4)^2\\
- &=\Bigl[\sqrt{(x+3)^2}+\sqrt{(2x+4)^2}\Bigr]\\
- &\quad\times\Bigl[\sqrt{(x+3)^2}-\sqrt{(2x+4)^2}\Bigr]\\
- &=(x+3+2x+4)(x+3-2x-4)\\
- &=(3x+7)(-x-1)
- \end{align*}
- \end{minipage}
- \newpage
- {\Large Exercice 28p142}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- T&=9x^2+12x+4\\
- &=(3x+2)^2
- \end{align*}
- \begin{align*}
- H&=x^2+169-26x\\
- &=(x-13)^2
- \end{align*}
- \begin{align*}
- A&=144x+144x^2+36\\
- &=(12x+6)
- \end{align*}
- \end{minipage}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- L&=49x^2+25-70x\\
- &=(7x-5)^2
- \end{align*}
- \begin{align*}
- E&=9x^2-24x+16\\
- &=(3x-4)^2
- \end{align*}
- \begin{align*}
- S&=-22x+121x^2+1\\
- &=(11x-1)^2
- \end{align*}
- \end{minipage}
- {\Large Exercice 29p142}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- P &=(6x-4)(2x+5)-(3x+2)(2x+5)\\
- &=(2x+5)\bigl[(6x-4)-(3x+2)\bigr]\\
- &=(2x+5)(6x-4-3x-2)\\
- &=(2x+5)(3x-6)
- \end{align*}
- \begin{align*}
- Y&=(4x-5)^2-(9-13x)^2\\
- &=(4x-5+9-13x)(4x-5-9+13x)\\
- &=(-9x+4)(17x-14)
- \end{align*}
- \end{minipage}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- T&=25x^2+9-30x\\
- &=(5x-3)^2
- \end{align*}
- \begin{align*}
- H&=(5x-7)(3x-2)-(x-8)(3x-2)\\
- &=(3x-2)(5x-7-x+8)\\
- &=(3x-2)(4x-1)
- \end{align*}
- \end{minipage}
- {\Large Exercice 30p142}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- M&=(4x-3)^2-28x+21\\
- &=(4x-3)^2-7(4x-3)\\
- &=(4x-3)(4x-3-7)\\
- &=(4x-3)(4x-10)
- \end{align*}
- \begin{align*}
- A&=5x-7+(7-5x)^2\\
- &=(7-5x)^2-(7-5x)\\
- &=(7-5x)(7-5x-1)\\
- &=(7-5x)(6-5x)
- \end{align*}
- \end{minipage}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- T&=-4x^2+20x-25+4x^2-25\\
- &=-50+20x\\
- &=-10(5-2x)
- \end{align*}
- \begin{align*}
- H&=(5x-3)^2-(3-2x)^2\\
- &=\bigl[ (5x-3)+(3-2x) \bigr]\bigl[ (5x-3)-(3-2x) \bigr]\\
- &=3x(7x-6)
- \end{align*}
- \end{minipage}
- {\Large Exercice 31p142}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- S&=(3x+2)(4x-1)+(8x-2)(7x-8)\\
- &=(3x+2)(4x-1)+2(4x-1)(7x-8)\\
- &=(4x-1)(3x+2+7x-8)\\
- &=(4x-1)(10x-6)
- \end{align*}
- \begin{align*}
- T&=(6x + 1)(3 - x) - 36x^2 + 1\\
- &=(6x + 1)(3 - x) - (6x+1)(6x-1)\\
- &=(6x+1)(3-x-6x+1)\\
- &=(6x+1)(4-7x)
- \end{align*}
- \begin{align*}
- R &= (4x + 4)(x - 5) + ( 3x + 3)( x - 9)\\
- &=4(x+1)(x-5)+3(x+1)(x-9)\\
- &=(x+1)\bigl[ 4(x-5)+3(x-9) \bigr]\\
- &=(x+1)(4x-20+3x-21)\\
- &=(x+1)(7x-41)
- \end{align*}
- \begin{align*}
- O&=(x+1)(7-x)-(-1-x)(x+7)\\
- &=(x+1)(7-x)+(x+1)(x+7)\\
- &=(x+1)(7-x+x+7)\\
- &=14(x+1)
- \end{align*}
- \end{minipage}
- \begin{minipage}[t]{6cm}
- \begin{align*}
- P&=x^2+6x+9-(2x-3)(x+3)\\
- &=(x+3)^2-(2x-3)(x+3)\\
- &=(x+3)(x+3-2x+3)\\
- &=(x+3)(6-x)
- \end{align*}
- \begin{align*}
- H&=(3x-4)(x+5)-(-6x+1)(x+5)\\
- &=(x+5)(3x-4+6x-1)\\
- &=(x+5)(9x-5)
- \end{align*}
- \begin{align*}
- E&=(8x-1)^2-(5x+2)(8x-1)\\
- &=(8x-1)(8x-1-5x-2)\\
- &=(8x-1)(3x-3)
- \end{align*}
- \end{minipage}
- {\Large Exercice 59p142}
- \begin{align*}
- 48^2-47^2-46^2+45^2&=\\
- 2304-2209-2116+2025&=4
- \end{align*}
- \begin{align*}
- 82^2-81^2-80^2+79^2&=\\
- 6724-6561-6400+6241&=4
- \end{align*}
- \begin{align*}
- 166^2-165^2-164^2+163^2&=\\
- 27556-27225-26896+26569&=4
- \end{align*}
- Le résulat de ce programme sera toujours égal à 4.
- \begin{align*}
- n&\in \mathbb{N} \\
- &\quad n^2-(n-1)^2-(n-2)^2+(n-3)^2\\
- &=n^2-(n^2-2n+1)-(n^2-4n+4)+(n^2-6n+9)\\
- &=n^2-n^2+2n-1-n^2+4n-4+n^2-6n+9\\
- &=9-5\\
- &=4
- \end{align*}
- Je n'ai jamais trouvé le même résultat au calcul :
- $$123456789515^2+123456789512^2-(123546879514^2+123456789513^2)$$
- Mais il devrait faire 4 :
- \begin{align*}
- &\quad n^2+(n-3)^2-\bigr[(n-1)^2+(n-2)^2\bigl]\\
- &=n^2+n^2-6n+9-(n^2-2n+1+n^2-4n+4)\\
- &=n^2+n^2-6n+9-n^2+2n-1-n^2+4n-4\\
- &=9-5\\
- &=4
- \end{align*}
- \end{document}
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