Section: 当领导系数值不等于 1 时乘数 | 5.2 主导系数值不等于1的乘数

Section outline

The area of a square is $\begin{align*}9x^2 + 24x + 16\end{align*}$ . What are the dimensions of the square?
::方形区域为 9x2+24x+16。方形的尺寸是多少?

Factoring Quadratic Functions
::计算二次函数

When the $\begin{align*}a\end{align*}$ term , the coefficient of the $\begin{align*}x^2\end{align*}$ term in a quadratic function in the form $\begin{align*}ax^2+bx+c,\end{align*}$ is greater than one, factoring it may be a bit tricky . This is because the $\begin{align*}a\end{align*}$ term is the result of multiplying the coefficients of the $\begin{align*}x\end{align*}$ terms from the source binomials, so you need to identify the different coefficients that are factors of $\begin{align*}a.\end{align*}$
::当一个词时, 方形 x2+bx+c 的二次函数中的 x2 术语的系数大于一个时, 乘以它可能有点困难。这是因为该词是源二元性乘以 x 术语的系数的结果, 所以您需要确定一个因素的不同系数。

As an introduction, consider FOIL-ing two binomials when the coefficients in front of the $\begin{align*}x\text{-}\end{align*}$ terms are not 1:
::作为导言,在xx期之前的系数不是1时,考虑FOIL确定两个二元论:

Multiply, using the FOIL method: $\begin{align*}(3x-5)(2x + 1)\end{align*}$ .
::乘数,使用FOIL方法: (3x-5)(2x+1)。

FIRST: $\begin{align*}3x \cdot 2x = 6x^2\end{align*}$
::第一: 3x% 2x=6x2

OUTSIDE: $\begin{align*}3x \cdot 1 = 3x\end{align*}$
::外部: 3x%1=3x

INSIDE: $\begin{align*}\text{-}5 \cdot 2x = \text{-}10x\end{align*}$
::内部: - 52x=- 10x

LAST: $\begin{align*}\text{-}5 \cdot 1 = \text{-}5\end{align*}$
::末期: -51=5

Combining all the terms together, you get: $\begin{align*}6x^2+3x-10x-5 = 6x^2-7x-5.\end{align*}$
::将所有术语合并后,您可以得到: 6x2+3x- 10x- 5= 6x2- 7x- 5。

Now, try working backwards and factor a trinomial with $\begin{align*}a>1\end{align*}$ to get two factors. Remember, you can always check your work by multiplying the final factors together.
::现在, 尝试向后工作, 并用一个 > 1 来乘以三边数, 以获得两个因素。记住, 您总是可以通过将最终因素乘以一起来检查您的工作。

Factor $\begin{align*}6x^2-x-2\end{align*}$
::6x2-x-2因数

This is a factorable trinomial. When there is a coefficient, or number in front of, $\begin{align*}x^2\end{align*}$ , you must follow all the steps you are familiar with to factor . M ultiply together $\begin{align*}a\end{align*}$ and $\begin{align*}c, \text{ from }ax^2+bx+c,\end{align*}$ and then find the two numbers whose product is $\begin{align*}ac\end{align*}$ and sum is $\begin{align*}b\end{align*}$ . One way to organize this information is to draw a large X with $\begin{align*}b\end{align*}$ in the top opening , and $\begin{align*}a \cdot c\end{align*}$ in the bottom one:
::这是一个可因数三角。当 x2 前面有一个系数或数字时, 您必须跟随您熟悉的参数的所有步骤。乘以 a 和 c, 从 x2+bx+c 乘以 a 和 c, 然后找到两个数字, 其产品为 ac 和和和 b。组织此信息的方法之一是在顶部开口绘制一个大 X 和 b , 在底部绘制 a 和 a :

For the side sections, you are looking for the two factors of $\begin{align*}a \cdot c,\end{align*}$ (12, in this case) that sum to be equal to $\begin{align*}b.\end{align*}$
::对于侧侧部分,您正在寻找ac(12,此处指12)等值于b的两个系数。

Factors	Sum
-1, 12	11
1, -12	-11
2, -6	-4
-2, 6	4
$\begin{align}{\color{red}3, \ \text{-}4}\end{align}$	$\begin{align}{\color{red}\text{-}1}\end{align}$
-3, 4	1

The factors that work are 3 and -4. Now, take these factors and rewrite the $\begin{align*}-x\end{align*}$ term, expanded as $\begin{align*}\text{-}4x+3x:\end{align*}$
::起作用的因素是 3 和 - 4。现在, 以这些因素来重写- x 术语, 扩展为 - 4x+3x :

\begin{aligned} 6 x^{2} - x - 2 \\ ↙ ↘ \\ 6 x^{2} - 4 x + 3 x - 2 \end{aligned}

$\begin{align*}& \quad \ 6x^2 {\color{red}-x}-2\\ & \qquad \ \ {\color{red} \swarrow}{\color{red} \searrow}\\ & 6x^2{\color{red}-4x+3x}-2\end{align*}$
::6x2-x-2 6x2-4x+3x-2

Next, group the first two terms together and the last two terms together and pull out any common factors.
::接下来,将前两个术语组合在一起,最后两个术语组合在一起,并找出任何共同因素。

\begin{aligned} (6 x^{2} - 4 x) + (3 x - 2) \\ 2 x (3 x - 2) + 1 (3 x - 2) \end{aligned}

$\begin{align*}& (6x^2-4x)+(3x-2)\\ & 2x(3x-2)+1(3x-2)\end{align*}$
:

6x2-4x)+(3x-2)2x(3x-2)+1(3x-2)

What is in the parenthesis is the same . We now have two terms that both have $\begin{align*}(3x-2)\end{align*}$ as factor. Pull this factor out.
::括号中的内容是一样的。我们现在有两个条件, 两者都具有( 3x-2) 系数。拔出这个系数。

The factors of $\begin{align*}6x^2-x-2\end{align*}$ are $\begin{align*}(3x-2)(2x + 1)\end{align*}$ . You can FOIL these to check your answer.
::6x2-x-2 的因数是 (3x-2)(2x+1) 。您可查看这些因数以查看您的答复。

Now, let's factor $\begin{align*}4x^2+8x-5\end{align*}$ .
::现在,让我们来考虑乘数4x2+8x-5。

Let’s make the steps we followed in the previous problem a little more concise.
::让我们让我们在前一个问题上所遵循的步骤更简洁一点。

Step 1: Find $\begin{align*}ac\end{align*}$ and the factors of this number that add up to $\begin{align*}b\end{align*}$ .
::第1步:查找 ac 和该数字中与 b 相加的系数。

$\begin{align*}4 \cdot -5=-20\end{align*}$ The factors of -20 that add up to 8 are 10 and -2.
::======================================================================================================================================================

Step 2: Rewrite the trinomial with the $\begin{align*}x-\end{align*}$ term expanded, using the two factors from Step 1.
::第2步:利用第1步的两个因素,以扩大的x-期重写三角关系。

\begin{aligned} 4 x^{2} + 8 x - 5 \\ ↙ ↘ \\ 4 x^{2} + 10 x - 2 x - 5 \end{aligned}

$\begin{align*}& \quad \ 4x^2+{\color{red}8x}-5\\ & \qquad \quad \ {\color{red} \swarrow}{\color{red} \searrow}\\ & 4x^2+{\color{red}10x-2x}-5\end{align*}$
::4x2+8x-5 4x2+10x-2x-5

Step 3: Group the first two and second two terms together, find the GCF and factor again.
::第3步:将前两个和第二个两个任期组合起来,再次发现绿色气候基金和因素。

\begin{aligned} (4 x^{2} + 10 x) + (- 2 x - 5) \\ 2 x (2 x + 5) - 1 (2 x + 5) \\ (2 x + 5) (2 x - 1) \end{aligned}

$\begin{align*}& (4x^2+10x)+(-2x-5)\\ & 2x(2x+5)-1(2x+5)\\ & (2x+5)(2x-1)\end{align*}$
:

4x2+10x)+(-2x-5)2x(2x+5)-2x-2x+5)-1(2x+5)(2x+5)(2x+5)(2x-1)

Alternate Method : What happens if we list $\begin{align*}-2x\end{align*}$ before $\begin{align*}10x\end{align*}$ in Step 2?
::替代方法:如果我们在步骤2中列出-2x在10x之前出现什么情况?

\begin{aligned} 4 x^{2} - 2 x + 10 x - 5 \\ (4 x^{2} - 2 x) (10 x - 5) \\ 2 x (2 x - 1) + 5 (2 x - 1) \\ (2 x - 1) (2 x + 5) \end{aligned}

$\begin{align*}& 4x^2-2x+10x-5\\ & (4x^2-2x)(10x-5)\\ & 2x(2x-1)+5(2x-1)\\ & (2x-1)(2x+5)\end{align*}$
::4x2-2x+10x-5(4x2-2-2x)(10x-5)2x(2x-1)+5(2x-1)-2x-1(2x-1)-1(2x-1)-2x-1(2x+5)

This tells us it does not matter which $\begin{align*}x-\end{align*}$ term we list first in Step 2 above.
::这告诉我们,我们在上文步骤2中首先列出哪个x-期并不重要。

Let's factor $\begin{align*}12x^2-22x-20\end{align*}$ .
::12x2 -22x -20乘以12x2 -22x -20

Let’s use the steps from the previous problem , but we are going to add an additional step at the beginning.
::让我们从上一个问题中走一步, 但我们在开始的时候还要再加一步。

Step 1: Look for any common factors. Pull out the GCF of all three terms, if there is one.
::第1步:寻找任何共同因素。如果存在一个条件,则取消全球合作框架的所有三个条件。

\begin{aligned} 12 x^{2} - 22 x - 20 = 2 (6 x^{2} - 11 x - 10) \end{aligned}

$\begin{align*}12x^2-22x-20 = 2(6x^2-11x-10)\end{align*}$
::12x2-22x-20=2(6x2-11x-10)

This will make it much easier for you to factor what is inside the parenthesis.
::这将使您更容易计算括号内的内容。

Step 2: Using what is inside the parenthesis, find $\begin{align*}ac\end{align*}$ and determine the factors that add up to $\begin{align*}b.\end{align*}$
::第2步:使用括号内的内容,找到 ac 并确定构成 b 的因素。

\begin{aligned} 6 \cdot - 10 = - 60 \to - 15 \cdot 4 = - 60, - 15 + 4 = - 11 \end{aligned}

$\begin{align*}6 \cdot -10 = -60 \ {\color{red}\rightarrow -15 \cdot 4 = -60}, \ -15+4=-11\end{align*}$

The factors of -60 that add up to -11 are -15 and 4.
::总计为 -11的 -60 系数为 -15 和 4。

Step 3: Rewrite the trinomial with the $\begin{align*}x\text{-}\end{align*}$ term expanded, using the two factors from Step 2.
::步骤3:利用从步骤2得出的两个因素,以扩大的x期重写三边协议。

\begin{aligned} 2 (6 x^{2} - 11 x - 10) \\ 2 (6 x^{2} - 15 x + 4 x - 10) \end{aligned}

$\begin{align*}& 2(6x^2{\color{red}-11x}-10)\\ & 2(6x^2 {\color{red}-15x+4x}-10)\end{align*}$
::2(6x2-11x-10)(2-6x2-15x+4x-10)

Step 4: Group the first two and second two terms together, find the GCF and factor again.
::第4步:将前两个和第二个两个任期组合起来,再次发现绿色气候基金和因素。

\begin{aligned} 2 (6 x^{2} - 15 x + 4 x - 10) \\ 2 [(6 x^{2} - 15 x) + (4 x - 10)] \\ 2 [3 x (2 x - 5) + 2 (2 x - 5)] \\ 2 (2 x - 5) (3 x + 2) \end{aligned}

$\begin{align*}& 2(6x^2-15x+4x-10)\\ & 2 \left[(6x^2-15x)+(4x-10)\right]\\ & 2 \left[ 3x(2x-5)+2(2x-5)\right]\\ & 2(2x-5)(3x+2)\end{align*}$
::2(6x2-15x+4x-10)2[(6x2-15x)+(4x-10)]2[3x(2x-2x-5)+2(2x-2x-5)]2 [2x-5(3x+2)

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the dimensions of the square.
::早些时候,你被要求找到广场的方位

The dimensions of a square are its length and its width, so we need to factor the area $\begin{align*}9x^2 + 24x + 16\end{align*}$ .
::方形的尺寸是其长度和宽度, 所以我们需要将区域 9x2+24x+16 乘以。

We need to multiply together $\begin{align*}a\end{align*}$ and $\begin{align*}c\end{align*}$ (from $\begin{align*}ax^2+bx+c\end{align*}$ ) and then find the two numbers whose product is $\begin{align*}ac\end{align*}$ and whose sum is $\begin{align*}b\end{align*}$ .
::我们需要将a和c(从x2+bx+c)乘以1和c(从x2+bx+c),然后找到其产品为ac的两个数字,其总和为b。

Now we can see that we need the two factors of 144 that also add up to 24. Testing the possibilities, we find that $\begin{align*}12 \cdot 12 = 144\end{align*}$ and $\begin{align*}12 + 12 = 24\end{align*}$ .
::现在我们可以看到,我们需要144这两个因素,这些因素加起来达到24个。测试可能性,我们发现1212=144和12+12=24。

Now, take these factors and rewrite the $\begin{align*}x-\end{align*}$ term expanded using 12 and 12.
::采用这些因素,重写使用12和12扩大的x-期。

\begin{aligned} 9 x^{2} + 24 x + 16 \\ ↙ ↘ \\ 9 x^{2} + 12 x + 12 x + 16 \end{aligned}

$\begin{align*}& \quad \ 9x^2 {\color{red}+24x}+16\\ & \qquad \ \ {\color{red} \swarrow}{\color{red} \searrow}\\ & 9x^2{\color{red}+12x+12x}+16\end{align*}$
::9x2+24x+16 9x2+12x+12x+16

\begin{aligned} (9 x^{2} + 12 x) + (12 x + 16) \\ 3 x (3 x + 4) + 4 (3 x + 4) \end{aligned}

$\begin{align*}& (9x^2+12x)+(12x+16)\\ & 3x(3x+4)+4(3x+4)\end{align*}$
:

9x2+12x)+(12x+16)3x(3x+4)+4(3x+4)

We now have two terms that both have $\begin{align*}(3x+4)\end{align*}$ as factor. Pull this factor out.
::我们现在有两个条件,两个条件都以(3x+4)为因数,拉出这一因数。

The factors of $\begin{align*}9x^2+24x+16\end{align*}$ are $\begin{align*}(3x + 4)(3x + 4)\end{align*}$ , which are also the dimensions of the square.
::9x2+24x+16的因数是(3x+4)(3x+4),这些因数也是方形的尺寸。

Example 2
::例2

Multiply $\begin{align*}(4x-3)(3x + 5)\end{align*}$ .
::乘以 (4x-3)(3x+5)。

FOIL: $\begin{align*}(4x-3)(3x + 5) = 12x^2 +20x-9x-15 = 12x^2+11x-15\end{align*}$
::FIL: (4x-3 (3x+5)= 12x2+20x-9x-15=12x2+11x-15)

Example 3
::例3

Factor the following quadratic, if possible.
::如果可能,则以下列二次方位因数为因数。

$\begin{align*}15x^2-4x-3\end{align*}$
::15x2 - 4x - 3

Use the steps from the examples above. There is no GCF, so we can find the factors of $\begin{align*}ac\end{align*}$ that add up to $\begin{align*}b\end{align*}$ .
::使用上述例子中的步骤。没有绿色气候基金, 所以我们可以找到 ac 的系数, 加到 b 。

$\begin{align*}15 \cdot -3 = -45 \end{align*}$ The factors of -45 that add up to -4 are -9 and 5.
::15345 乘以 -4的-45系数为 -9和5。

\begin{aligned} 15 x^{2} - 4 x - 3 \\ (15 x^{2} - 9 x) + (5 x - 3) \\ 3 x (5 x - 3) + 1 (5 x - 3) \\ (5 x - 3) (3 x + 1) \end{aligned}

$\begin{align*}& 15x^2 {\color{red}-4x}-3\\ & (15x^2 {\color{red}-9x)}+({\color{red}5x}-3)\\ & 3x(5x-3)+1(5x-3)\\ & (5x-3)(3x+1)\end{align*}$
::15x2-4x-3(15x2-9x)+(5x-3)3x(5x-3)+1(5x-3)(5x-3)(3x+1)

Example 4
::例4

Factor the following quadratic, if possible.
::如果可能,则以下列二次方位因数为因数。

$\begin{align*}3x^2+6x-12\end{align*}$
::3x2+6x-12

$\begin{align*}3x^2+6x-12\end{align*}$ has a GCF of 3. Pulling this out, we have $\begin{align*}3(x^2+2x-6)\end{align*}$ . There is no number in front of $\begin{align*}x^2\end{align*}$ , so we see if there are any factors of -6 that add up to 2. There are not, so this trinomial is not factorable.
::3x2+6x-12全球合作框架为3。拔出这个系数,我们有3(x2+2x-6)x2前面没有数字,所以我们看看有没有6-6因素加起来等于2。没有,因此三重因素是不可考虑的。

Example 5
::例5

Factor the following quadratic, if possible.
::如果可能,则以下列二次方位因数为因数。

$\begin{align*}24x^2-30x-9\end{align*}$
::24x2-30x-9

$\begin{align*}24x^2-30x-9\end{align*}$ also has a GCF of 3. Pulling this out, we have $\begin{align*}3(8x^2-10x-3)\end{align*}$ . $\begin{align*}ac = -24\end{align*}$ . The factors of -24 than add up to -10 are -12 and 2.
::24x2-30x-9也有3个全球合作框架。解决这个问题,我们有3(8x2-10x-3) ac24。与-10相加的24个因素是 -12和2。

\begin{aligned} 3 (8 x^{2} - 10 x - 3) \\ 3 [(8 x^{2} - 12 x) + (2 x - 3)] \\ 3 [4 x (2 x - 3) + 1 (2 x - 3)] \\ 3 (2 x - 3) (4 x + 1) \end{aligned}

$\begin{align*}& 3(8x^2{\color{red}-10x}-3)\\ & 3 \left[(8x^2{\color{red}-12x})+({\color{red}2x}-3)\right]\\ & 3 \left[ 4x(2x-3)+1(2x-3)\right]\\ & 3(2x-3)(4x+1)\end{align*}$
::3(3)8x2-10x-3)3[(8x2-12x)+(2x-3)]3[(4x(2x-3)+1(2x-3)]3(2x-3)(4x+1)

Example 6
::例6

Factor the following quadratic, if possible.
::如果可能,则以下列二次方位因数为因数。

$\begin{align*}4x^2+4x-48\end{align*}$
::4x2+4x-48

$\begin{align*}4x^2+4x-48\end{align*}$ . has a GCF of 4. Pulling this out, we have $\begin{align*}4(x^2+x-12)\end{align*}$ . This trinomial does not have a number in front of $\begin{align*}x^2\end{align*}$ , so we can use the shortcut we are familiar with . What are the factors of -12 that add up to 1?
::4x2+4x- 48。 4x2+4x- 48。全球合作框架为4:拉出来, 我们有 4(x2+x-12) 12。这个三角体在 x2 前面没有数字, 所以我们可以使用我们熟悉的捷径。 12 因素加到 1 是什么?

\begin{aligned} 4 (x^{2} + x - 12) \\ 4 (x + 4) (x - 3) \end{aligned}

$\begin{align*}& 4(x^2+x-12)\\ & 4(x+4)(x-3)\end{align*}$
::4(x2+x-12)4(x+4)(x-3)

Review
::回顾

Multiply the following expressions.
::乘以下列表达式。

$\begin{align*}(2x-1)(x + 5)\end{align*}$
:2x-1(x+5))

$\begin{align*}(3x + 2)(2x-3)\end{align*}$
:3x+2)(2x-3)

$\begin{align*}(4x + 1)(4x-1)\end{align*}$
:4x+1)(4x-1)

Factor the following quadratic equations, if possible. If they cannot be factored, write not factorable . Don’t forget to look for any GCFs first.
::如果可能的话, 则以下列二次方程为因数。如果无法计算的话, 请写不可计算。不要忘记先寻找任何 GFSP 。

$\begin{align*}5x^2+18x+9\end{align*}$
::5x2+18x+9

$\begin{align*}6x^2-21x\end{align*}$
::6x2 - 21x

$\begin{align*}10x^2-x-3\end{align*}$
::10x2-x-3

$\begin{align*}3x^2+2x-8\end{align*}$
::3x2+2x-8

$\begin{align*}4x^2+8x+3\end{align*}$
::4x2+8x+3

$\begin{align*}12x^2-12x-18\end{align*}$
::12x2-12x-18

$\begin{align*}16x^2-6x-1\end{align*}$
::16x2-6x-1

$\begin{align*}5x^2-35x+60\end{align*}$
::5x2-35x+60

$\begin{align*}2x^2+7x+3\end{align*}$
::2x2+7x+3

$\begin{align*}3x^2+3x+27\end{align*}$
::3x2+3x+27

$\begin{align*}8x^2-14x-4\end{align*}$
::8x2 - 14x - 4

$\begin{align*}10x^2+27x-9\end{align*}$
::10x2+27x-9

$\begin{align*}4x^2+12x+9\end{align*}$
::4x2+12x+9

$\begin{align*}15x^2+35x\end{align*}$
::15x2+35x

$\begin{align*}6x^2-19x+15\end{align*}$
::6x2-19x+15

Factor $\begin{align*}x^2-25\end{align*}$ . What is $\begin{align*}b\end{align*}$ ?
::乘以 x2 - 25。什么是 b?

Factor $\begin{align*}9x^2-16\end{align*}$ . What is $\begin{align*}b\end{align*}$ ? What types of numbers are $\begin{align*}a\end{align*}$ and $\begin{align*}c\end{align*}$ ?
::9x2-16。b是什么?a和c是哪种数字?

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

Factoring Quadratic Functions ::计算二次函数

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)

Factoring Quadratic Functions
::计算二次函数

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Review
::回顾

Review (Answers)
::回顾(答复)