a/

\(a\left(b-c\right)-b\left(a+c\right)+c\left(a-b\right)=\)

\(=ab-ac-ab-bc+ac-bc=-2bc\)

b/

\(a\left(1-b\right)+a\left(a^2-1\right)=\)

\(=a-ab+a^3-a=a^3-ab=a\left(a^2-b\right)\)

c/

\(a\left(b-x\right)+x\left(a+b\right)=ab-ax+ax+bx=\)

\(=ab+bx=b\left(a+x\right)\)

Ta đặt

  \(A=1\times3+3\times5+...+61\times63\)

\(6A=1\times3\times6+3\times5\times6+....+61\times63\times6\)

\(6A=1\times3\times6+3\times5\times\left(7-1\right)+...+61\times63\times\left(65-59\right)\)

\(6A=1\times3\times6+3\times5\times7-1\times3\times5+...+61\times63\times65-59\times61\times63\)

\(6A=1\times3\times6-1\times3\times5+61\times63\times65\)

\(6A=3+61\times63\times65\)

\(6A=3\times\left(1+61\times21\times65\right)\)

\(2A=83266\)

\(A=83266\div2=41633\)

\(a,x\left(y-z\right)+y\left(z-x\right)+z\left(x-y\right)\\ =xy-xz+yz-xy+xz-yz\\ =\left(xy-xy\right)+\left(xz-xz\right)+\left(yz-yz\right)\\ =0+0+0\\ =0\left(dpcm\right)\)

\(b,x\left(y+z-yz\right)-y\left(z+x-zx\right)+z\left(y-x\right)\\ =xy+xz-xyz-yz-xy+xyz+yz-xz\\ =\left(xy-xy\right)+\left(xz-xz\right)+\left(xyz-xyz\right)+\left(yz-yz\right)\\ =0+0+0+0\\ =0\left(dpcm\right)\)

a) 

\(P=\left(x^{14}-9x^{13}\right)-\left(x^{13}-9x^{12}\right)+\left(x^{12}-9x^{11}\right)-...+\left(x^2-9x\right)-\left(x-9\right)+1\)

\(=x^{13}\left(x-9\right)-x^{12}\left(x-9\right)+x^{11}\left(x-9\right)+...+x\left(x-9\right)-\left(x-9\right)+1\)

\(P\left(9\right)=1\)

b)

\(Q=\left(x^{15}-7x^{14}\right)-\left(x^{14}-7x^{13}\right)+\left(x^{13}-7x^{12}\right)-...-\left(x^2-7x\right)+\left(x-7\right)+2\)

\(=x^{14}\left(x-7\right)-x^{13}\left(x-7\right)+x^{12}\left(x-7\right)-...-x\left(x-7\right)+\left(x-7\right)+2\)

\(Q\left(7\right)=2\)

b) Ta có:

 P(x) + H(x) = x⁴ - x³ + 2x² + x + 1

=> H(x) = x⁴ - x³ + 2x² + x + 1 - P(x)

=> H(x) = (x⁴ - x³ + 2x² + x + 1) - (2x⁴ - x² + x - 2)

=> H(x) = -x⁴ - x³ + 3x² + 3

Vậy H(x) = -x⁴ - x³ + 3x² + 3

Bạn nên ghi đề bằng công thức toán (biểu tượng $\sum$ góc trái khung soạn thảo) để được hỗ trợ tốt hơn. Viết như thế này khá khó đọc.

\(\left(3b^2\right)^2-b^3.\left(1-5b\right)\)

\(=9b^4-b^3+5b^4\)

\(=14b^4-b^3\)

4x\(^{1+n-2}\)y\(^{1+n+1}\)4xy\(^{1+n}\)+4xy