Demandes associées au chapitre-notebook: "Manipulations algébriques"

Pour accéder au code relatif à un exemple, double-cliquer sur le triangle du crochet icônique à droite.

Exemple 1

Expand[-1 + (((1 - z))^5) - z]

Expand[-1 + (((1 - z))^5) - z] /. z -> a + I b

Expand[-1 + (((1 - z))^5) - z /. z -> a + I b]

Exemple 2

Factor[
      (((((y^2)/2) - (1 + 6 (t^2))))^2) + (((y + 8 (t^3)))^2) - 
        (((1 + 4 (t^2)))^3)]

Exemple 3

\(TraditionalForm
    \`Together[
      1\/\(x\^6 + 33\ x\^4 + 56\ x\^2 - 20\ x - 61\ x\^3 - 9\ x\^5\) + 
        1\/\(x\^5 + 23\ x\^3 + 12\ x - 28\ x\^2 - 8\ x\^4\)]\)

1/Factor[(x^5) - 8 (x^4) + 23 (x^3) - 28 (x^2) + 12 x] + 
      1/Factor[(x^6) - 9 (x^5) + 33 (x^4) - 61 (x^3) + 56 (x^2) - 20 x]

PolynomialLCM[
      (-3 + x) (((-2 + x))^2) (-1 + x) x, 
      (((-2 + x))^2) (-1 + x) x (5 - 4 x + (x^2))]

(-3 + x) + (5 - 4 x + (x^2))

Factor[%]

Collect[
      (-3 + x) (((-2 + x))^2) (-1 + y x) x 
          (5 - 4 x y + (x^2)) + (x^6) - 10 (x^5) + 39 (x^4) - 74 (x^3) + 
        68 (x^2) - 24 x, x]

Collect[
      (-3 + x) (((-2 + x))^2) (-1 + y x) x 
          (5 - 4 x y + (x^2)) + (x^6) - 10 (x^5) + 39 (x^4) - 74 (x^3) + 
        68 (x^2) - 24 x, y]

?Apart

Apart[
      (((x^4) + 7 (x^3) + 23 (x^2) + 35 x + 22)/((((1 + x))^2) 
          ((x^2) + 4 x + 7)))]

PolynomialDivision[22 + 35 x + 23 (x^2) + 7 (x^3) + (x^4), 
      Expand[(((1 + x))^2) (7 + 4 x + (x^2))], x]

donnée = 
      ((15 + 17 x + 7 (x^2) + (x^3))/((((1 + x))^2) 
          (7 + 4 x + (x^2))))

fraction = 
      Together[(A/(((x + 1))^2)) + (B/(x + 1)) + 
          ((C x + D)/((x^2) + 4 x + 7))]

Simplify[(((x + 1))^2) fraction] /. x -> -1

Simplify[fraction /. x -> 0] /. A -> 1

donnée /. x -> 0 /. A -> 1

Simplify[fraction /. x -> 1] /. A -> 1

donnée /. x -> 1 /. A -> 1

Simplify[fraction /. x -> 2] /. A -> 1

donnée /. x -> 2 /. A -> 1

Solve[{1 + B + (D/7) == (15/7), (1/12) (3 + 6 B + C + D) == (5/6), 
        (1/9) + (B/3) + (C/19) + (D/19) == (85/171)}, {B, C, D}]

1 + (A/(((1 + x))^2)) + (B/(1 + x)) + 
        ((C x + D)/(7 + 4 x + (x^2))) /. {A -> 1, C -> 0, 
        B -> 1, D -> 1}

E1 = 
      -(((-5 + 2 x) (3 + 2 x))/(((5 - 5 x + (x^2)))^2))
                 + (2/(5 - 5 x + (x^2)))

Expand[%]

Apart[%%]

Simplify[%]

E2 = ((6 a (x^5))/(Sqrt[(a^2) + (b^2)]))

TableForm[Table[E2, {a, -2, 2}, {b, n, n + 2}], 
      TableSpacing -> {2, 4}]

Options[TableForm]

TableForm[Table[E2, {a, -2, 2}, {b, n, n + 2}], 
      TableSpacing -> {5, 4}]

TableForm[Table[E2, {a, -2, 2}, {b, n, n + 2}], 
      TableSpacing -> {1, 3}]

MatrixForm[Table[E2, {a, -2, 2}, {b, n, n + 2}], 
      TableSpacing -> {1, 3}]

E3 = 
      ((4 a (x^3))/(Sqrt[(a^2) + (x^2)])) - 
        ((x (b + a (x^4)))/((((a^2) + (x^2)))^(3/2)))

Expand[%]

Simplify[%]

\(A = Apart[%\/\@\(a\^2 + x\^2\)]\)

B = % /. {a -> b}

Solve[(b^5) - b == 0]

B /. %

With[{solutions = Solve[(b^5) - b == 0]}, 
      MapThread[Print, {b /. solutions, B /. solutions}]];

"pour b = "-1
          " on a B = "0
        ";"

"pour b = "#1
          " on a B = "#2&

With[{solutions = Solve[(b^5) - b == 0]},MapThread["pour b = "#1
                  " on a:  B = "#2
                  ";"&,{b /. solutions, B /. solutions}]]

TableForm[%]

(x - a) (x - (a^2)) (x - (a^3)) (x - (a^4))

Expand[%]

expression = %

expression /. a -> 1

expression /. a -> -(1/3)

Factor [%]

Exemple 4

Collect[
      (-3 + x) (((-2 + x))^2) (-1 + y x) x 
          (5 - 4 x y + (x^2)) + (x^6) - 10 (x^5) + 39 (x^4) - 74 (x^3) + 
        68 (x^2) - 24 x, x]

Collect[
      (-3 + x) (((-2 + x))^2) (-1 + y x) x 
          (5 - 4 x y + (x^2)) + (x^6) - 10 (x^5) + 39 (x^4) - 74 (x^3) + 
        68 (x^2) - 24 x, y]

Exemple 5

Apart[
      (((x^4) + 7 (x^3) + 23 (x^2) + 35 x + 22)/((((1 + x))^2) 
          ((x^2) + 4 x + 7)))]

Exemple 6

E1 = 
      -(((-5 + 2 x) (3 + 2 x))/(((5 - 5 x + (x^2)))^2))
                 + (2/(5 - 5 x + (x^2)))

Expand[%]

Apart[%%]

Simplify[%]

Exemple 7

E2 = ((6 a (x^5))/(Sqrt[(a^2) + (b^2)]))

TableForm[Table[E2, {a, -2, 2}, {b, n, n + 2}], 
      TableSpacing -> {2, 4}]

TableForm[Table[E2, {a, -2, 2}, {b, n, n + 2}], 
      TableSpacing -> {5, 4}]

TableForm[Table[E2, {a, -2, 2}, {b, n, n + 2}], 
      TableSpacing -> {1, 3}]

MatrixForm[Table[E2, {a, -2, 2}, {b, n, n + 2}], 
      TableSpacing -> {1, 3}]

Exemple 8

E3 = 
      ((4 a (x^3))/(Sqrt[(a^2) + (x^2)])) - 
        ((x (b + a (x^4)))/((((a^2) + (x^2)))^(3/2)))

Expand[%]

Simplify[%]

With[{solutions = Solve[(b^5) - b == 0]},MapThread["pour b = "#1
                  " on a:  B = "#2
                  ";"&,{b /. solutions, B /. solutions}]]

Exemple 9

expression /. a -> 1

expression /. a -> Log[2]

expression /. a -> Exp[((2 I π)/5)]

Simplify[%]

Collect[%, x]

{Re[(((-1))^(3/5))], Im[(((-1))^(3/5))]}

Simplify[expression /. a -> Exp[((2 I π)/5)]]

CoefficientList[
      Simplify[expression /. a -> Exp[((2 I π)/5)]], x]

coefficient = {Re[%], Im[%]}

Simplify [%]

Table [Exp[((2  I n π)/5)],  {n,  1,  4}]

{Re[%], Im[%]}

MapThread [List,  %]

\(Show\ [Graphics\ [Map\ [{AbsolutePointSize\ [5], Point\ [#]}&, \ %]], 

      	Axes\  -> \ True, \ AspectRatio\  -> \ Automatic]\)