本文将教你如何因式分解二次多项式。一个多项式含有一个变量(x),x有特定的次数,多项式还有各种其他的变量和常数。要因式分解一个二次多项式成多个多项式因子相乘的形式,你的数学水平得达到代数I以上,否则不太容易理解本方法的原理。
本文中都用到的标准形式的二次多项式:ax2 + bx + c = 0
步骤
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1
写下表达式。 以次数高低排列,如果有最大公因数则提出来:6 + 6x2 + 13x,6x2 + 13x + 6
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2用以下方法之一,得出因式分解的结果:(2x + 3)(3x + 2)
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3
用FOIL(首项相乘、外项相乘、内向相乘、次项相乘,这是展开多项式相乘的一种步骤方法)分解,并合并同类项:(2x + 3)(3x + 2),6x2 + 4x + 9x + 6,6x2 + 13x + 6。
方法 1 的 6:
试错法
若你的多项式十分简单,可以自己来发现因数。注意:用这个方法,可能不能因式分解更复杂的三项式了。
例子: 3x2 + 2x - 8
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1
把a、c的因数写出来:a = 3 因数有: 1 和 3,c = -8 因数: 2 和 4 和 1 和 8
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2写两对括号,留点空白:( x )( x )
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3把a可能的一对因数写在x前:本例子中只有一对因数 (3 x )(1 x )
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4在x项后面分别写上成对的c的因数,先试试 (3x 8 )(x 1 )
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5
决定x项和常数项的符号。 以下是方法:如果ax2 + bx + c 则 (x + h)(x + k),如果 ax2 - bx - c 或 ax2 + bx - c 则 (x - h)(x + k)。如果 ax2 - bx + c 则 (x - h)(x - k)。本例子中是 3x2 + 2x - 8 ,因此 (x - h)(x + k)是答案的形式,然后试试: (3x + 8)(x - 1)
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6
把两个括号展开,如果中间项不对,则这种化简不对(c的因数选错了)。 (3x + 8)(x - 1),3x2 - 3x + 8x - 8,3x2 + 5x - 8 ≠ 3x2 + 2x - 8
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7
如果必要,则换掉因数。 本例中我们试试2和4这对: (3x + 2)(x - 4)
c 现在是-8。
但是外项和内项积分别是-12x 和 2x, 合并不成+2x。
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8
如果必要的话就调转顺序。 我们试试把2、4换个位置。 (3x + 4)(x - 2)
c 还是对的。
外项积和内项积是-6x 和 4x, 则这两个数的和同2x正好符号相反
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9
然后再确认一下符号正负。 顺序是没错的,现在把符号倒过来: (3x - 4)(x + 2)
c 还是对的。
外项积和内项积现在6x 和 -4x。 加起来等于2x ,这次就对了。
方法 2 的 6:
分解法
不喜欢猜的方法, 可以试试这个。
例子: 6x2 + 13x + 6
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1把a、c乘起来,本例中是:6•6 = 36
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2找出一对数字,乘起来是36,加起来又是b(13):4•9 = 36 4 + 9 = 13
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3
把两个数字设为 k 和 h (顺序随意): ax2 + kx + hx + c,6x2 + 4x + 9x + 6
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4
整理成组,因式分解。 整理一下方程,使得可以提出最大公因式((3x+2)),然后合并同类项,得到因式分解结果。6x2 + 4x + 9x + 6,2x(3x + 2) + 3(3x + 2),(2x + 3)(3x + 2)
方法 3 的 6:
三重方法
本方法很像分解法,不过更简单
例子: 8x2 + 10x + 2
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1
将a、c两项相乘。 8•2 = 16
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2
找出两个数字,相乘是16,相加又是b(10)。 2•8 = 16 8 + 2 = 10
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3将两个数( h 、 k )代入这个方程:(ax + h)(ax + k)---------------------- a(8x + 8)(8x + 2)---------------------- 8(如图)
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4
看看哪一个括号项可以被a整除,并且商是偶数。 a {本例中为(8x + 8)}。用a 除以这个数,让另一项保持原样(8x + 8)(8x + 2)---------------------- 8,答案:(x + 1)(8x + 2)
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5如果两括号有最大公因式,提出来:(x + 1)(8x + 2),2(x + 1)(4x + 1)
方法 4 的 6:
两个平方之差
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1
如果需要,则提出最大公因数。 27x2 - 12,3(9x2 - 4)
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2
看看方程是否是两个平方之差。 一定要有两项,否则不能平均分解这个方程。√(9x2 ) = 3x , √(4) = 2 (注意这里省去了负数根。)
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3把“a”、“c”从你的等式中代入下列公式:(√(a) + √(c))(√(a) - √(c))3[(√(9x2 ) + √(4))(√(9x2 ) - √(4))]3[(3x + 2)(3x - 2)]
方法
5
方法 5 的 6:
使用二次公式
上述方法都不行,则用二次公式
例如:x2 + 4x + 1
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1将对应量代入本方程:x = -b ± √(b2 - 4ac) --------------------- 2a,x = -4 ± √(42 - 4•1•1) ----------------------- 2•1(如图)
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2
解出x。 得到两个x,x= -4 ± √(16 - 4) ------------------ 2x = -4 ± √(12) -------------- 2x = -4 ± √(4•3) -------------- 2x = -4 ± 2√(3) -------------- 2x = -2 ± √(3),x = -2 + √(3) 或 x = -2 - √(3)(如图)
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3把x值(h 、k ) 代入方程 (x - h)(x - k),(x - (-2 + √(3))(x - (-2 - √(3)),(x + 2 + √(3))(x + 2 - √(3))
方法
6
方法 6 的 6:
用计算器
这些步骤适合TI图形计算器,在标准考试中尤其好用。
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1输入[Y = ] :y = x2 − x − 2
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2
按下 [GRAPH]作图。
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3
找到和x 轴相交点得到(-1, 0), (2 , 0),x = -1, x = 2
如果看不到,则按下[2nd] -[TRACE], 按下 [2] 或选择“0”。移到交点之左以后按下[ENTER], 移到交点之右按下[ENTER], 移到尽量接近和x轴相交的点旁边,按下 [ENTER],计算器就会自动算出该点的横坐标。对另一个交点也重复此步骤。
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4
把x值(h 和 k )代入本公式: (x - h)(x - k),(x - (-1))(x - 2),整理为(x + 1)(x + (-2)) 表示出两个交点来。
利用箱型法(可视解)
本网站有解释: http://www.purplemath.com/modules/factquad3.htm
视频说明: http://www.youtube.com/watch?v=bq1Iw1w1Bgo
小提示
若用二次公式因式分解了一个多项式,其中含有根数,可能需要将x换成分数来检查该解是否正确。
如果一个项没有系数,则系数是1。x2 = 1x2
如果有 TI-84 计算器 (可画图) ,则有一个叫做SOLVER的程序可以解二次方程,这个程序还可以解任何其他次数的多项式。
如果一个项不存在,则它的系数是0。有时把0项写出来会比较方便,比如x2 + 6 = x2 + 0x + 6
在熟悉试错法前,先把要试的因数写下来,熟练了以后再在脑子中运算。
警告
如果你在数学课中学到了这个概念,要注意老师建议用什么方式,就尽量不要用你喜欢的别的方式。因为老师可能会让你在考试中用特定的一种方法来解,或者不让你用图画计算器来解。
你需要准备
铅笔
纸
二次方程,或叫二次多项式
画图计算器(可选)