Toán Lớp 8: Tìm đa thức M a) x^3-5x^2+x-5=(x-5).M b) (x^4-x^3-4x^2-5x-3):M=x^2+x+1

Question

Toán Lớp 8:  Tìm đa thức M a) x^3-5x^2+x-5=(x-5).M b) (x^4-x^3-4x^2-5x-3):M=x^2+x+1

tonhu12 · Answer

$ text{a) x&sup3; - 5x&sup2; + x - 5 = (x - 5)M}$   $ text{&hArr; M = $ dfrac{x&sup3; - 5x&sup2; + x - 5}{x - 5}$ $=$ $ dfrac{x^2 (x - 5) + (x - 5)}{x - 5}$ $=$ $ dfrac{(x - 5)(x^2 + 1)}{x - 5}$ $=$ $x^2 + 1$}$   $ text{KL: Vậy M = $x^2 + 1$}$   $ text{b) ($x^4$ - x&sup3; - 4x&sup2; - 5x - 3) : M = x&sup2; + x + 1}$   $ text{&hArr; $x^4$ - x&sup3; - 4x&sup2; - 5x - 3 = (x&sup2; + x + 1).M}$   $ text{&hArr; $x^4$ + x&sup3; + x&sup2; - 2x&sup3; - 2x&sup2; - 2x - 3x - 3x&sup2; - 3 = M(x&sup2; + x + 1)}$   $ text{&hArr; x&sup2;(x&sup2; + x + 1) - 2x(x&sup2; + x + 1) - 3(x&sup2; + x + 1) = M(x&sup2; + x + 1)}$   $ text{&hArr; (x&sup2; + x + 1)(x&sup2; - 2x - 3 = M(x&sup2; + x + 1)}$ $  $   $ text{&hArr; M = $ dfrac{(x&sup2; + x + 1)(x&sup2; - 2x - 3)}{x&sup2; + x + 1}$ $=$ $x^2 - 2x - 3$ }$ $  $ $ text{KL: M = $x^2 $ - 2x - 3}$    @tryphena

thanhtung · Answer

#tnvt a)x^3-5x^2+x-5=(x-5).M <=>x^2(x-5)+(x-5)=M.(x-5) <=>(x-5)(x^2+1)=M.(x-5) ->M=x^2+1 $ $ b)(x^4-x^3-4x^2-5x-3):M=x^2+x+1 <=>x^4-x^3-4x^2-5x-3=M.(x^2+x+1) <=>x^4+x^3+x^2-2x^3-2x^2-2x-3x^2-3x-3=M.(x^2+x+1) <=>x^2(x^2+x+1)-2x(x^2+x+1)-3(x^2+x+1)=M.(x^2+x+1) <=>(x^2+x+1)(x^2-2x-3)=M.(x^2+x+1) =>M=x^2-2x-3 $ $