1,求X的值a)X^3-6X^2+11X-6=0b)X^4-10X^3+35X^2-50X+24=0c)X^5-15X^4+85X^3-225X^2+274X-120=02,determineandexplainpatternsintheroots.意思就是确定并且解释是什么模式在这个根中3,conjecture(hypothesize)cluesintheequationthat

Question

1,求X的值
a)X^3-6X^2+11X-6=0
b)X^4-10X^3+35X^2-50X+24=0
c)X^5-15X^4+85X^3-225X^2+274X-120=0
2,determine and explain patterns in the roots.
意思就是确定并且解释是什么模式在这个根中
3,conjecture(hypothesize) clues in the equation that can help you predict patterns in the roots.
4,test your conjecture on the following equations:
a)X^3-7X+6=0
b)X^4-X^3-7X^2+13X-6=0
c)X^6-21X^5+175X^4+1624X^2-1764X+720=0

Answer 1

1,求X的值
a)X^3-6X^2+11X-6=0
x^3-6x^2+9x+2x-6=0
x(x-3)^2+2(x-3)=0
(x-3)(x^2-3x+2)=0
x1=3,x2=1,x3=2(三个不等实根）
b)X^4-10X^3+35X^2-50X+24=0
x^4-x^3-9x^3+9x^2+26x^2-26x-24x+24=0
x^3(x-1)-9x^2(x-1)+26x(x-1)-24(x-1)=0
(x-1)(x^3-9x^2+26x-24)=0
(x-1)[x^2(x-2)-7x(x-2)+12(x-2)]=0
(x-1)(x-2)(x^2-7x+12)=0
x1=1,x2=2,x3=3,x4=3
c)X^5-15X^4+85X^3-225X^2+274X-120=0
仿上面的方法,换一种写知
令f(x)=x^5-15x^4+85x^3-225x^2+274x-120
则f(1)=1-15+85-225+274-120=0,故x1=1
x^5-15x^4+85x^3-225x^2+274x-120=(x-1)(x^4-14x^3+71x^2-154x+120)
同理,x=2时,x^4-14x^3+71x^2-154x+120=0
x^4-14x^3+71x^2-154x+120=（x-2)(x^3-12x^2+47x-60)
x^3-12x^2+47x-60=（x-3)(x-9x+27)
所以x1=1,x2=2,x3=3,x4,5=(9+/-3根号3倍i)/2(三实根,两虚根）
4,仿第一题做法知
a)X^3-7X+6=0
（x-1)(x^2+x-6)=0
(x-1)(x-2)(x+3)=0
x1=1,x2=2,x3=-3
b)X^4-X^3-7X^2+13X-6=0
(x-1)(x^3-7x+6)=0
x1=x2=1,x3=2,x4=-3
c)X^6-21X^5+175X^4+1624X^2-1764X+720=0(应该还有-735x^3吧）
(x-1)(x^5-20x^4+155x^3-580x^2+1044x-720)=0
(x-1)(x-2)(x^4-18x^3+119x^2-342x+360)=0
(x-1)(x-2)(x-3)(x^3-15x^2+74x-120)=0
(x-1)(x-2)(x-3)(x-4)(x^2-11x+30)=0
(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=0
x1=1,x2=2,x3=3,x4=4,x5=5,x6=6
(以上用计算机更方便）