Section: 零产品原则 | 9.7 零产品原则

Section outline

Zero Product Principle
::零产品原则

The most useful thing about factoring is that we can use it to help solve polynomial equations.
::考虑因素方面最有用的是我们可以用它来帮助解决多面方程式。

Solving for X
::X 的解答

Consider an equation like $\begin{aligned} 2 x^{2} + 5 x - 42 = 0 \end{aligned}$ . How do you solve for $\begin{aligned} x \end{aligned}$ ?
::考虑像 2x2+5x- 42=0 这样的方程式。 x 如何解析 ?

There’s no good way to isolate $\begin{aligned} x \end{aligned}$ in this equation, so we can’t solve it using any of the techniques we’ve already learned. But the left-hand side of the equation can be factored, making the equation $\begin{aligned} (x + 6) (2 x - 7) = 0 \end{aligned}$ .
::在这个方程式中,没有把 x 分离出来的好方法,所以我们无法使用我们已经学到的任何技术来解决它。但方程式的左侧可以被计算,使方程式(x+6)(2x-7)=0。

How is this helpful? The answer lies in a useful property of multiplication : if two numbers multiply to zero, then at least one of those numbers must be zero. This is called the Zero-Product Property.
::这有用吗?答案在于一个有用的乘法属性:如果两个数字乘以零,那么其中至少一个数字必须是零。这叫做零产品属性。

What does this mean for our polynomial equation? Since the product equals 0, then at least one of the factors on the left-hand side must equal zero. So we can find the two solutions by setting each factor equal to zero and solving each equation separately.
::这对我们的多边等式意味着什么?既然产品等于0,那么左侧至少有一个因素必须等于零。这样我们就可以通过将每个因素设定为零和分别解决每个等式来找到两个解决方案。

Setting the factors equal to zero gives us:
::设定等于零的因数给我们带来:

$\begin{aligned} (x + 6) = 0 & OR & (2 x - 7) = 0 \end{aligned}$

:x+6)=0OR(2x-7)=0

Solving both of those equations gives us:
::解决这两个方程式告诉我们:

$\begin{aligned} x + 6 = 0 & 2 x - 7 = 0 \\ \underline{\underline{x = - 6}} & OR & 2 x = 7 \\ \underline{\underline{x = \frac{7}{2}}} \end{aligned}$

::x+6=02x-7=0x____OR2x=7x=72__

Notice that the solution is $\begin{aligned} x = - 6 \end{aligned}$ OR $\begin{aligned} x = \frac{7}{2} \end{aligned}$ . The OR means that either of these values of $\begin{aligned} x \end{aligned}$ would make the product of the two factors equal to zero. Let’s plug the solutions back into the equation and check that this is correct.
::请注意解决方案是 x6ORx=72 。 OR 意味着这些 x 的任何一个值都会使这两个系数的产值等于零。让我们将解决方案插回方程式中, 检查是否正确。

$\begin{aligned} C h e c k : x = - 6; & C h e c k : x = \frac{7}{2} \\ (x + 6) (2 x - 7) = & (x + 6) (2 x - 7) = \\ (- 6 + 6) (2 (- 6) - 7) = & (\frac{7}{2} + 6) (2 \cdot \frac{7}{2} - 7) = \\ (0) (- 19) = 0 & (\frac{19}{2}) (7 - 7) = \\ (\frac{19}{2}) (0) = 0 \end{aligned}$

::检查: x% 6; 检查: x=72( x+6)( 2x-7) = (x+6)( 2x-7) = (x+6)( 2x-7) = (- 6+6)( 2) - 7) = (72+6)( 2( - 6) = (2) - 72 - 7) = (0)( - 19) = 0( 192)( 7) = (192)(0) = 0

Both solutions check out.
::两种解决方案都检查了出来。

Factoring a polynomial is very useful because the Zero-Product Property allows us to break up the problem into simpler separate steps. When we can’t factor a polynomial, the problem becomes harder and we must use other methods that you will learn later.
::将多元性因素考虑在内非常有用,因为零产品属性允许我们将问题分成更简单的单独步骤。当我们无法将多元性因素考虑在内时,问题就变得更加严重了,我们必须使用你以后会学到的其他方法。

As a last note in this section, keep in mind that the Zero-Product Property only works when a product equals zero. For example, if you multiplied two numbers and the answer was nine, that wouldn’t mean that one or both of the numbers must be nine. In order to use the property, the factored polynomial must be equal to zero.
::作为本节的最后一个注释,请记住“零产品属性”只有在产品等于零时才起作用。比如,如果乘以两个数字,答案是9,那么这并不意味着数字中的一个或两个都必须是9。为了使用该财产,计算系数的多元性必须等于零。

Solving for an Unknown Value
::解决未知值

Solve each equation:
::解决每个方程式:

Since all the are in factored form , we can just set each factor equal to zero and solve the simpler equations separately
::由于所有因素都以因素化的形式出现,我们可以将每个因素设定为零,分别解决简单的方程式

a) $\begin{aligned} (x - 9) (3 x + 4) = 0 \end{aligned}$
::a) (x-9) (3x+4)=0

$\begin{aligned} (x - 9) (3 x + 4) = 0 \end{aligned}$ can be split up into two linear equations:
:x- 9) (3x+4)=0 可以分为两个线性方程:

$\begin{aligned} x - 9 = 0 & 3 x + 4 = 0 \\ \underline{\underline{x = 9}} & or & 3 x = - 4 \\ \underline{\underline{x = - \frac{4}{3}}} \end{aligned}$

::x- 9=03x+4=0x=9_ or3x4x43_

b) $\begin{aligned} x (5 x - 4) = 0 \end{aligned}$
:b) x(5x-4)=0

$\begin{aligned} x (5 x - 4) = 0 \end{aligned}$ can be split up into two linear equations:
::x( 5x- 4) =0 可分为两个线性方程 :

$\begin{aligned} 5 x - 4 = 0 \\ \underline{\underline{x = 0}} & or & 5 x = 4 \\ \underline{\underline{x = \frac{4}{5}}} \end{aligned}$

::5x-4=0x=0__或5x=4x=45__

c) $\begin{aligned} 4 x (x + 6) (4 x - 9) = 0 \end{aligned}$
:c) 4x(x+6)(4x-9)=0

$\begin{aligned} 4 x (x + 6) (4 x - 9) = 0 \end{aligned}$ can be split up into three linear equations:
::4x(x+6)(4x-9)=0可分为三个线性方程:

$\begin{aligned} 4 x = 0 & x + 6 = 0 & 4 x - 9 = 0 \\ x = \frac{0}{4} & or & x = - 6 & or & 4 x = 9 \\ \underline{\underline{x = 0}} & \underline{\underline{x = \frac{9}{4}}} \end{aligned}$

::4x=0x+6=04x-9=0x=04orx=%6or4x=9x0__x=94__

Solve Simple Polynomial Equations by Factoring
::以保理方式解决简单聚合等同

Now that we know the basics of factoring, we can solve some simple polynomial equations. We already saw how we can use the Zero-Product Property to solve polynomials in factored form—now we can use that knowledge to solve polynomials by factoring them first. Here are the steps:
::现在我们知道保理学的基本原理了, 我们可以解决一些简单的多元方程式。我们已经看到了我们如何用零生产属性解决因子化的多生产形式—现在我们可以用这种知识解决多生产方程式,先用保理法解决多生产方程式。下面是步骤:

a) If necessary, rewrite the equation in standard form so that the right-hand side equals zero.
::a) 必要时,以标准格式重写方程式,使右侧等于零。

b) Factor the polynomial completely.
:b) 将多元性完全乘以。

c) Use the zero-product rule to set each factor equal to zero.
:c) 使用零产品规则,将每个系数设定为零。

d) Solve each equation from step 3.
:d) 从第3步解决每个方程。

e) Check your answers by substituting your solutions into the original equation
::e) 以原来的方程代替自己的方程,检查答案

Solving Polynomial Equations
::溶解聚合等同

Solve the following polynomial equations.
::解决以下多面方程式。

a) $\begin{aligned} x^{2} - 2 x = 0 \end{aligned}$
::a) x2-2x=0

$\begin{aligned} x^{2} - 2 x = 0 \end{aligned}$
::x2-2x=0

Rewrite: this is not necessary since the equation is in the correct form.
::重写: 这没有必要, 因为方程式的形状是正确的。

Factor: The common factor is $\begin{aligned} x \end{aligned}$ , so this factors as $\begin{aligned} x (x - 2) = 0 \end{aligned}$ .
::系数:共同系数为 x,因此该系数为 x(x-2)=0。

Set each factor equal to zero:
::设定等于零的因数 :

$\begin{aligned} x = 0 & or & x - 2 = 0 \end{aligned}$

::x=0orx-2=0

Solve:
::解决 :

$\begin{aligned} \underline{x = 0} & or & \underline{x = 2} \end{aligned}$

::x=0_orx=2_

Check: Substitute each solution back into the original equation.
::复选: 将每个溶液替换回原始方程。

$\begin{aligned} x & = 0 \Rightarrow (0)^{2} - 2 (0) = 0 & works out \\ x & = 2 \Rightarrow (2)^{2} - 2 (2) = 4 - 4 = 0 & works out \end{aligned}$

::x=0(0)2-2(0)=0works outx=2(2)2-2(2)=4-4=0works outx=2(2)(2)=4-4=0

Answer: $\begin{aligned} x = 0, x = 2 \end{aligned}$
::答复:x=0,x=2

b) $\begin{aligned} 2 x^{2} = 5 x \end{aligned}$
::b) 2x2=5x

$\begin{aligned} 2 x^{2} = 5 x \end{aligned}$
::2x2=5x

Rewrite: $\begin{aligned} 2 x^{2} = 5 x \Rightarrow 2 x^{2} - 5 x = 0 \end{aligned}$
::重写: 2x2=5x%2x2- 5x=0

Factor: The common factor is $\begin{aligned} x \end{aligned}$ , so this factors as $\begin{aligned} x (2 x - 5) = 0 \end{aligned}$ .
::系数:共同系数为 x,因此该系数为 x(2x-5)=0。

Set each factor equal to zero:
::设定等于零的因数 :

$\begin{aligned} x = 0 & or & 2 x - 5 = 0 \end{aligned}$

::x=0or2x-5=0

Solve:
::解决 :

$\begin{aligned} \underline{x = 0} & or & 2 x = 5 \\ \underline{x = \frac{5}{2}} \end{aligned}$

::x=0_ or2x=5x=52_

Check: Substitute each solution back into the original equation.
::复选: 将每个溶液替换回原始方程。

$\begin{aligned} x & = 0 \Rightarrow 2 (0)^{2} = 5 (0) \Rightarrow 0 = 0 & works out \\ x & = \frac{5}{2} \Rightarrow 2 {(\frac{5}{2})}^{2} = 5 \cdot \frac{5}{2} \Rightarrow 2 \cdot \frac{25}{4} = \frac{25}{2} \Rightarrow \frac{25}{2} = \frac{25}{2} & works out \end{aligned}$

::x=0=02(0)2=5(0)0=0works outx=522(2(52)2=5522(22254=252}252=252=252 worksworksworksworksworkswork out

Answer: $\begin{aligned} x = 0, x = \frac{5}{2} \end{aligned}$
::答复:x=0,x=52

c) $\begin{aligned} 9 x^{2} y - 6 x y = 0 \end{aligned}$
:c) 9x2y-6xy=0

$\begin{aligned} 9 x^{2} y - 6 x y = 0 \end{aligned}$
::9x2y-6xy=0

Rewrite: not necessary
::重写: 不需要重写

Factor: The common factor is $\begin{aligned} 3 x y \end{aligned}$ , so this factors as $\begin{aligned} 3 x y (3 x - 2) = 0 \end{aligned}$ .
::系数:共同系数为 3xy, 3x( 3x-2)=0。

Set each factor equal to zero:
::设定等于零的因数 :

$\begin{aligned} 3 = 0 \end{aligned}$ is never true, so this part does not give a solution. The factors we have left give us:
::3=0永远不会是真实的,所以本部分没有给出解决方案。我们留下的因素给了我们:

$\begin{aligned} x = 0 & or & y = 0 & or & 3 x - 2 = 0 \end{aligned}$

::x=0ory=0or3x-2=0

Solve:
::解决 :

$\begin{aligned} \underline{x = 0} & or & \underline{y = 0} & or & 3 x = 2 \\ \underline{x = \frac{2}{3}} \end{aligned}$

::x=0_ory=0_or3x=2x=23_

Check: Substitute each solution back into the original equation.
::复选: 将每个溶液替换回原始方程。

$\begin{aligned} x = 0 \Rightarrow 9 (0) y - 6 (0) y = 0 - 0 = 0 & works out \\ y = 0 \Rightarrow 9 x^{2} (0) - 6 x (0) = 0 - 0 = 0 & works out \\ x = \frac{2}{3} \Rightarrow 9 \cdot {(\frac{2}{3})}^{2} y - 6 \cdot \frac{2}{3} y = 9 \cdot \frac{4}{9} y - 4 y = 4 y - 4 y = 0 & works out \end{aligned}$

::x=09( 0)y- 6( 0)y=0-0=0-0=0worksouty=09x2( 0)-6x( 0)=0-0=0works outx=239( 23) (23)-2y-623y=949y-4y=4y-4y=0

Answer: $\begin{aligned} x = 0, y = 0, x = \frac{2}{3} \end{aligned}$
::答复:x=0,y=0,x=23

Example
::示例示例示例示例

Example 1
::例1

Solve the following polynomial equation.
::解决以下的多元等式。

$\begin{aligned} 9 x^{2} - 3 x = 0 \end{aligned}$

::9x2 - 3x=0

Solution: $\begin{aligned} 9 x^{2} - 3 x = 0 \end{aligned}$
::解析度: 9x2- 3x=0

Rewrite : This is not necessary since the equation is in the correct form.
::重写: 这没有必要, 因为方程式的形状是正确的。

Factor : The common factor is $\begin{aligned} 3 x \end{aligned}$ , so this factors as: $\begin{aligned} 3 x (3 x - 1) = 0 \end{aligned}$ .
::系数: 共同系数为 3x3, 因此此系数为 3x( 3x-1) = 0 。

Set each factor equal to zero.
::设定每个系数等于零。

$\begin{aligned} 3 x = 0 or 3 x - 1 = 0 \end{aligned}$

::3x=0 或 3x-1=0

Solve :
::解决 :

$\begin{aligned} x = 0 or x = \frac{1}{3} \end{aligned}$

::x=0 或 x=13 x=13

Check : Substitute each solution back into the original equation.
::复选: 将每个溶液替换回原始方程。

$\begin{aligned} x = 0 : & 9 (0)^{2} - 3 (0) = 0 \\ x = \frac{1}{3} : & 9 {(\frac{1}{3})}^{2} - 3 (\frac{1}{3}) = 0 \\ 9 (\frac{1}{9}) - 3 (\frac{1}{3}) = 0 \\ 1 - 1 = 0 \end{aligned}$

::x=0:9(0)2-3(0)=0x=13:9(13)2-3(13)=0(13)-9(19)-3(13)=0-9(19)-3(13)=0-1-1=0

Answer $\begin{aligned} x = 0, x = \frac{1}{3} \end{aligned}$
::答案=0, x=13

Review
::回顾

Solve the following polynomial equations.
::解决以下多面方程式。

$\begin{aligned} x (x + 12) = 0 \end{aligned}$
:xx+12)=0

$\begin{aligned} (2 x + 1) (2 x - 1) = 0 \end{aligned}$
:2x+1)(2x-1)=0

$\begin{aligned} (x - 5) (2 x + 7) (3 x - 4) = 0 \end{aligned}$
:x-5)(2x+7)(3x-4)=0

$\begin{aligned} 2 x (x + 9) (7 x - 20) = 0 \end{aligned}$
::2xx+9(7x-20)=0

$\begin{aligned} x (3 + y) = 0 \end{aligned}$
:x3+y)=0

$\begin{aligned} x (x - 2 y) = 0 \end{aligned}$
:x--2y)=0

$\begin{aligned} 18 y - 3 y^{2} = 0 \end{aligned}$
::18-3y2=0

$\begin{aligned} 9 x^{2} = 27 x \end{aligned}$
::9x2=27x

$\begin{aligned} 4 a^{2} + a = 0 \end{aligned}$
::4a2+a=0

$\begin{aligned} b^{2} - \frac{5}{3} b = 0 \end{aligned}$
::b2-53b=0

$\begin{aligned} 4 x^{2} = 36 \end{aligned}$
::4x2=36

$\begin{aligned} x^{3} - 5 x^{2} = 0 \end{aligned}$
::x3 - 5x2=0

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

Zero Product Principle ::零产品原则

Solving for X ::X 的解答

Solving for an Unknown Value ::解决未知值

Solve Simple Polynomial Equations by Factoring ::以保理方式解决简单聚合等同

Solving Polynomial Equations ::溶解聚合等同

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Zero Product Principle
::零产品原则

Solving for X
::X 的解答

Solving for an Unknown Value
::解决未知值

Solve Simple Polynomial Equations by Factoring
::以保理方式解决简单聚合等同

Solving Polynomial Equations
::溶解聚合等同

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)