Question

因式分解，求解，急

a³＋b³＋3(a²＋b²）＋3（a＋b）＋2
3x³－7x＋10
x的4次方－4x＋3
x³－11x²＋31x－21
2x³－5x²＋1

Answer 1

(1) a³+b³+3(a²+b²)+3(a+b)+2
=(a³+3a²+3a+1)+(b³+3b²+3b+1) (两个括号内分别用完全立方公式)
=(a+1)³+(b+1)³ (再用立方和公式)
=(a+b+2)[(a+1)²+(a+1)(b+1)+(b+1)²]

(2) 3x³-7x+10
=(3x³+6x²)-(6x²+12x)+(5x+10)
=3x²(x+2)-6x(x+2)+5(x+2)
=(3x²-6x+5)(x+2) (实数范围内不能再分解)

(3) x^4-4x+3
=(x^4-x)-(3x-3)
=x(x³-1)-3(x-1) (第一个括号内用立方差公式)
=x(x-1)(x²+x+1)-3(x-1)
=(x-1)(x³+x²+x-3)
=(x-1)[(x³-1)+(x²-1)+(x-1)]
=(x-1)[(x-1)(x²+x+1)+(x-1)(x+1)+(x-1)]
=(x-1)²[(x²+x+1)+(x+1)+1]
=(x-1)²(x²+2x+3)

(4) x³－11x²＋31x－21
=(x³-x²)-(10x²-10x)+(21x-21)
=x²(x-1)-10x(x-1)+21(x-1)
=(x²-10x+21)(x-1)
=(x-1)(x-3)(x-7)

(5) 2x³－5x²＋1
=(2x³-x²)-(4x²-1) (后一个括号用平方差公式)
=x²(2x-1)-(2x+1)(2x-1)
=(2x-1)(x²-2x-1) (有理数范围内不能继续分解)
=(2x-1)(x-1-根号2)(x-1+根号2)