math -factorisation－lgzrelv的部落格

標題:

math -factorisation

發問:

factorise (1)a^2(b+c)+b^2(c-a)+c^2(b-a)-2abc (2)x^2(b-c)+b^2(c+x)-c^2(x+b) (3)x^2(b^2-c^2)+b(c^2-x^2)+c(x^2-b^2) (4)x^2(y+z)+y^2(z-x)+z^2(y-x)-2xyz

最佳解答:

(1) a2(b + c) + b2(c - a) + c2(b - a) - 2abc = a2b + a2c + b2c - ab2 + bc2 - ac2 - 2abc = a2b - 2abc + bc2 + a2c - ac2 - ab2 + b2c = b(a2 - 2ac + c2) + ac(a - c) - b2(a - c) = b(a - c)2 + ac(a - c) - b2(a - c) = (a - c)[ b(a - c) + ac - b2 ] = (a - c)(ab - bc + ac - b2) = (a - c)(ab - b2 + ac - bc) = (a - c)[ b(a - b) + c(a - b)] = (a - c)(a - b)(b + c) (2) x2(b - c) + b2(c + x) - c2(x + b) = x2(b - c) + b2c + b2x - c2x - bc2 = x2(b - c) + b2c - bc2 + b2x - c2x = x2(b - c) + bc(b -c) + x (b2 - c2) = x2(b - c) + bc(b -c) + x (b - c)(b + c) = (b - c)[x2 + bc + x(b+c)] = (b - c)(x2 + bc + bx + cx) = (b - c)(cx + x2 + bc + bx) = (b - c)[x(c + x) + b(c + x)] = (b - c)(c + x)( b + x) (3) x2(b2-c2)+b(c2-x2)+c(x2-b2) = x2(b + c)(b - c) + bc2 - bx2 + cx2 - b2c = x2(b + c)(b - c) - bx2 + cx2 - b2c + bc2 = x2(b + c)(b - c) - x2 (b - c) - bc(b - c) = (b - c) [x2(b + c) - x2 - bc] = (b - c) [x2(b + c -1) - bc] (4) x2(y+z)+y2(z-x)+z2(y-x)-2xyz = x2y + x2z + y2z - xy2 + yz2 - xz2 - 2xyz = x2y - 2xyz + yz2 + x2z - xz2 - xy2 + y2z = y(x2 - 2xz + z2) + xz(x - z) - y2(x - z) = y(x - z)2 + xz(x - z) - y2(x - z) = (x - z)[ y(x - z) + xz - y2 ] = (x - z)(xy - yz + xz - y2) = (x - z)(xy - y2 + xz - yz) = (x - z)[ y(x - y) + z(x - y)] = (x - z)(x - y)(y + z)

其他解答:

2) (b-c)x^2 + (b^2 - c^2)x + (b^2 c - b c^2) = (b-c)x^2 + (b-c)(b+c)x + bc(b-c) = (b-c)[x^2 + (b+c)x + bc] = (b-c)(x+b)(x+c) 做住一條,唔得閒..