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分解二次表达式:完全平方

学习如何分解"完全平方"形式的二次表达式。例如,将x²+6x+9写成(x+3)².
对多项式进行因式分解需要我们将多项式写成两个或多个多项式的乘积.这是多项式乘法的相反过程.
在本文中,我们将学习如何利用特殊规律来因式分解完全平方三项式。这个方法是 二项式的平方 的逆向运算, 所以请先熟知后者的运算方法。

介绍:分解完全平方三项式

为了展开任何二项式,我们可以运用一下公式的其中一个.
  • left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared, equals, start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared
  • left parenthesis, start color #11accd, a, end color #11accd, minus, start color #1fab54, b, end color #1fab54, right parenthesis, squared, equals, start color #11accd, a, end color #11accd, squared, minus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared
请注意在这些公式中,ab可以被替代为任何的代数式.比如,假设我们想要拓展left parenthesis, x, plus, 5, right parenthesis, squared.在这里,start color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accdstart color #1fab54, b, end color #1fab54, equals, start color #1fab54, 5, end color #1fab54,因此我们可以得出:
(x+5)2=x2+2(x)(5)+(5)2=x2+10x+25\begin{aligned}(\blueD x+\greenD 5)^2&=\blueD x^2+2(\blueD x)(\greenD5)+(\greenD 5)^2\\\\ &=x^2+10x+25\end{aligned}
你可以通过运用乘法来展开left parenthesis, x, plus, 5, right parenthesis, squared以检查这个公式是否正确.
这个展开过程的相反步骤就是 一种因式分解 .如果我们重新以相反顺序写下这个公式,我们就会得到分解多项式的公式,即a, squared, plus minus, 2, a, b, plus, b, squared.
  • start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared
  • start color #11accd, a, end color #11accd, squared, minus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, minus, start color #1fab54, b, end color #1fab54, right parenthesis, squared
我们可以运用这个公式来分解 x, squared, plus, 10, x, plus, 25. 那么我们就会得到start color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accdstart color #1fab54, b, end color #1fab54, equals, start color #1fab54, 5, end color #1fab54.
x2+10x+25=x2+2(x)(5)+(5)2=(x+5)2\begin{aligned}x^2+10x+25&=\blueD x^2+2(\blueD x)(\greenD5)+(\greenD 5)^2\\\\ &=(\blueD x+\greenD 5)^2\end{aligned}
这种形式的表达式被称为完全平方三项式.这个名称说明了这种三项式可以被表达为一个完全平方.
让我们看看一些我们可以运用这个公式来分解完全平方三项式的例子.

例题 1: 因式分解 x, squared, plus, 8, x, plus, 16

请注意第一项和最后一项都是完全平方. x, squared, equals, left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, squared16, equals, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, squared. 另外,请注意中间项是这两个被平方的数据的乘积的两倍:2, left parenthesis, start color #11accd, x, end color #11accd, right parenthesis, left parenthesis, start color #1fab54, 4, end color #1fab54, right parenthesis, equals, 8, x.
这告诉了我们如果多项式是完全平方三项式,我们我们就可以运用如下的公式来进行因式分解.
start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared
在这个例子中,start color #11accd, a, end color #11accd, equals, start color #11accd, x, end color #11accdstart color #1fab54, b, end color #1fab54, equals, start color #1fab54, 4, end color #1fab54.我们可以用如下方法进行因式分解:
x2+8x+16=(x)2+2(x)(4)+(4)2=(x+4)2\begin{aligned}x^2+8x+16&=(\blueD x)^2+2(\blueD x)(\greenD 4)+(\greenD4)^2\\ \\ &=(\blueD{x}+\greenD{4})^2\end{aligned}
我们可以通过拓展left parenthesis, x, plus, 4, right parenthesis, squared:来检验答案.
(x+4)2=(x)2+2(x)(4)+(4)2=x2+8x+16\begin{aligned}(x+4)^2&=(x)^2+2(x)(4)+(4)^2\\ \\ &=x^2+8x+16 \end{aligned}

看看你对知识掌握得如何

1) 因式分解 x, squared, plus, 6, x, plus, 9.
选出正确答案:
选出正确答案:

2) 因式分解 x, squared, minus, 6, x, plus, 9.
选出正确答案:
选出正确答案:

3) 因式分解 x, squared, plus, 14, x, plus, 49.

例题 2: 因式分解 4, x, squared, plus, 12, x, plus, 9

完全平方三项式的系数不一定要是1.
举例来说,在4, x, squared, plus, 12, x, plus, 9中,第一项和最后一项都是完全平方: 4, x, squared, equals, left parenthesis, start color #11accd, 2, x, end color #11accd, right parenthesis, squared9, equals, left parenthesis, start color #1fab54, 3, end color #1fab54, right parenthesis, squared. 除此之外,请注意中间项是被平方的两个数据的乘积的两倍:2, left parenthesis, start color #11accd, 2, x, end color #11accd, right parenthesis, left parenthesis, start color #1fab54, 3, end color #1fab54, right parenthesis, equals, 12, x.
因为这满足了上述的条件,4, x, squared, plus, 12, x, plus, 9 也是一个完全平方三项式. 我们可以在此运用如下的因式分解公式:
start color #11accd, a, end color #11accd, squared, plus, 2, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, plus, start color #1fab54, b, end color #1fab54, squared, space, equals, left parenthesis, start color #11accd, a, end color #11accd, plus, start color #1fab54, b, end color #1fab54, right parenthesis, squared
在这个例子中,start color #11accd, a, end color #11accd, equals, start color #11accd, 2, x, end color #11accdstart color #1fab54, b, end color #1fab54, equals, start color #1fab54, 3, end color #1fab54. 这个多项式的分解过程如下:
4x2+12x+9=(2x)2+2(2x)(3)+(3)2=(2x+3)2\begin{aligned}4x^2+12x+9&=(\blueD {2x})^2+2(\blueD {2x})(\greenD 3)+(\greenD3)^2\\ \\ &=(\blueD{2x}+\greenD{3})^2\end{aligned}
我们可以通过拓展left parenthesis, 2, x, plus, 3, right parenthesis, squared来检查答案.

检查你对本课的理解

4) 因式分解 9, x, squared, plus, 30, x, plus, 25.
选出正确答案:
选出正确答案:

5) 因式分解 4, x, squared, minus, 20, x, plus, 25.

挑战题

6*) 因式分解 x, start superscript, 4, end superscript, plus, 2, x, squared, plus, 1.

7*) 因式分解 9, x, squared, plus, 24, x, y, plus, 16, y, squared.

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