高校问答老k翻盘河池:因式分解
来源：百度文库 编辑：高校问答 时间：2024/04/30 15:27:41
(本内容作为C,D水平,供优秀生选用)
1. 半径为145的圆面积比半径为144的圆的面积大多少?(用π表示)
2.求证:(n2+3n+1)2-1是连续四个整数的积,(其中n为整数)
3.已知:a+b=1,求证:a3+b3+3ab=1.
4.已知:5x2-2xy+y2+4x+1=0,求x,y的值.
5.求证:不论x,y是怎样的正数,多项式x5+x4y-2x3y2-x2y3+2y5-xy4的值都是非负数.
6.三角形ABC的三边是a,b,c满足a2+b2+c2=ab+ac+ab,试判断三角形ABC的形状.
7.已知:a+b=-3,ab=-4,求多项式a2+a2b+ab2+b2的值.
8.已知:(a+b)(x+y)=2(ax+by),求证:a=b或x=y.
9.已知:a+b=c,试证明:c2-2bc+b2-a2=0.
10.设有理数xy满足x2+4xy-2x-4y+2=0,求x2y+y2x的值.
11.用简便方法计算:4×0.6252+4×0.3752.
1.
π(145^2-144^2)=π(145+144)(145-144)=289π
2.
(n^2+3n+1)^2-1
=(n^2+3n+2)(n^2+3n)
=(n+1)(n+2)n(n+3)
=n(n+1)(n+2)(n+3)
3.
a+b=1
(a+b)^3=a^3+3a^b+3ab^2+b^3=a^3+b^3+3ab(a+b)=a^3+b^3+3ab=1
证毕
4.
5x2-2xy+y2+4x+1
=(y-x)^2+(4x^2+4x+1)
=(y-x)^2+(2x+1)^2
=0
y=x,2x+1=0
则x=y=-1/2
5.
x5+x4y-2x3y2-x2y3+2y5-xy4
=(2y^5-2x^3y^2)+(x^5-x^2y^3)+(x^4y-xy^4)
=-2y^2(x^3-y^3)+x^2(x^3-y^3)+xy(x^3-y^3)
=(x^3-y^3)(x^2+xy-2y^2)
=(x-y)(x^2+xy+y^2)(x-y)(x+2y)
=(x-y)^2(x+2y)[(x+1/2y)^2+3/4y^2]
(x-y)^2>=0,x+2y>0,(x+1/2y)^2+3/4y^2>=0
不论x,y是怎样的正数,多项式x5+x4y-2x3y2-x2y3+2y5-xy4的值都是非负数.
证毕
6.
a2+b2+c2=ab+ac+ab
2a^2+2b^2+2c^2-2ab-2ac-2bc=0
(a-b)^2+(a-c)^2+(b-c)^2=0
a=b=c,等边三角形
7.
a2+a2b+ab2+b2
=(a^2+b^2)+ab(a+b)
=(a+b)^2-2ab+ab(a+b)
=9+8+12
=29
8.
(a+b)(x+y)-2(ax+by)
=ax+ay+bx+by-2ax-2by
=bx-by+ay-ax
=b(x-y)+a(y-x)
=(b-a)(x-y)
=0
则a=b或x=y
证毕
9.
c2-2bc+b2-a2
=(b+c)^2-a^2
=(b+c-a)(b+c+a)
=0
题目中有错误,应该是b+c=0
10.
x2+4y2-2x-4y+2
=(x-1)^2+(2y-1)^2
=0
x=1,y=1/2
x2y+y2x=xy(x+y)=1*1/2*(1+1/2)=3/4
11.
4×0.6252+4×0.3752
=4*(0.625+0.0002)+4*(0.375+0.0002)
=4*(0.625+0.375)+4*(0.0002+0.0002)
=4*1+0.0016
=4.0016
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