节: 由单式和FOIL方法乘法 | 6.4 单体和FIL方法乘法

章节大纲

You are asked to frame a picture. You want the width of the frame to be 5 inches longer than the width of the glass and the height of the frame to be 7 inches longer than the height of the glass. You measure the glass and find the height to width ratio is 4:3. By multiplying the length and the width, you can write a polynomial to describe the area that the picture in the frame will cover. In this section, we begin to discuss multiplying .
::您被要求设置图片框。您希望框架宽度比玻璃宽度长5英寸,框架高度比玻璃高度长7英寸。您测量了玻璃, 发现其高度与宽度之比是 4: 3 。乘以长度和宽度, 您可以写一个多符号来描述框中图片所覆盖的区域。在本节中, 我们开始讨论乘法。

Multiplying a Polynomial by a Monomial
::以单声道乘以多声道

To multiply a polynomial by a monomial , there are a couple of previous concepts that we need to recall.
::要将多元性乘以单一性,我们需要回顾过去的一些概念。

Example 1
::例1

Simplify $\begin{aligned} (5 x^{3}) (- 8 x^{7}) \end{aligned}$ .
::简化( 5x3)(- 8x7) 。

Solution: To multiply a monomial by a monomial, we need to recall the product rule of exponents : $\begin{aligned} b^{m} \cdot b^{n} = b^{m + n} \end{aligned}$ . Then, we multiply numbers with numbers and variables with variables.
::解决方案 : 要将一个单式乘以一个单式, 我们需要记住推手的产物规则 : bmbn=bm+n 。然后, 我们用数字和变量来乘以数字和变量。

$\begin{aligned} (5 x^{3}) (- 8 x^{7}) = (5 \cdot - 8) (x^{3} \cdot x^{7}) = - 40 x^{(3 + 7)} = - 40 x^{10} \end{aligned}$

:5x3 (- 8x7) = (58 (x3x7) = 40x( 3+7) 40x10)

WARNING
::警告

It is a common mistake to multiply the exponents: $\begin{aligned} x^{3} \cdot x^{7} \neq x^{3 \cdot 7} \neq x^{21} \end{aligned}$ .
::乘以指数是一个常见错误: x3x7x3x77x21。

If we expand each of the terms , we can see why we need to add NOT multiply the exponents. The first term is $\begin{aligned} x^{3} = x \cdot x \cdot x \end{aligned}$ , so we are multiplying three x 's together. The second term is $\begin{aligned} x^{7} = x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \end{aligned}$ where we are multiplying seven x 's together. All together the total number of x 's is ten x 's, or the sum of 3 and 7.
::如果我们对每个术语进行扩展, 我们可以看到为什么我们需要添加不乘以前例。第一个术语是 x3=xxxxxxxx, 所以我们将三个 x 相乘。第二个术语是 x7=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx, 我们乘以七个 x 共乘以七个 x 共乘。 x 共共共乘十 x 或三和七。

Example 2
::例2

Find $\begin{aligned} 2 x (x^{2} + 4) \end{aligned}$ .
::查找 2x( x2+4) 。

Solution: Recall the that said $\begin{aligned} a (b + c) = a b + a c \end{aligned}$ . Here, we want to distribute the 2 x to each of the terms in the " data-term="Parentheses" role="term" tabindex="0"> parentheses .
::解答: 回顾 a(b+c) =ab+ac 所说的 a(b+c) =ab+ac。在此, 我们要将 2x 分配给括号中的每个词。

$\begin{aligned} 2 x (x^{2} + 4) = (2 x \cdot x^{2}) + (2 x \cdot 4) = 2 x^{3} + 8 x \end{aligned}$

::2x(xx2+4) =( 2xxxx2) +( 2x4) = 2x3+8x

We can expand this idea to polynomials with more terms.
::我们可以用更多的条件把这个想法推广到多教派

by CK-12 demonstrates how to multiply a monomial and a polynomial.
::由 CK-12 演示如何乘以单式和多式。

Example 3
::例3

Find $\begin{aligned} - 2 x^{2} (3 x^{3} - 4 x^{2} + 12 x - 9) \end{aligned}$ .
::查找 - 2x2( 3x3- 4x2+12x- 9) 。

Solution: Use the distributive property to multiply $\begin{aligned} - 2 x^{2} \end{aligned}$ by the polynomial.
::解决方案: 将分配属性乘以 2- 2x2 乘以多数值。

$\begin{aligned} - 2 x^{2} (3 x^{3} - 4 x^{2} + 12 x - 9) & = (- 2 x^{2} \cdot 3 x^{3}) + (- 2 x^{2} \cdot - 4 x^{2}) + (- 2 x^{2} \cdot 12 x) + (- 2 x^{2} \cdot - 9) \\ = - 6 x^{5} + 8 x^{4} - 24 x^{3} + 18 x^{2} \end{aligned}$

::-2x2(3x3-4x2+12x-9)=(-2x2x2x3x3)+(-2x2x24x2)+(-2x2x2x12x)+(-2x2x2x12x)+(-2x2x2*9)\*6x5+8x4-24x3+18x2

WARNING
::警告

Make sure to distribute to all of the terms in the polynomial not just the first one.
::确保将多纪念书中的所有术语, 不仅仅是第一个术语。

For example, $\begin{aligned} - 2 x^{2} (3 x^{3} - 4 x^{2} + 12 x - 9) \neq - 6 x^{5} - 4 x^{2} + 12 x - 9 \end{aligned}$ .
::例如,-2x2(3x3-4x2+12x-9)6x5-4x2+12x-9。

FOIL Method
::东帝汶方法

FOIL is a mnemonic device used to organize your thinking while multiplying binomials by binomials. To , we need to distribute each of the terms in the first binomial to the second binomial.
::FOIL 是一个记忆设备, 用来组织您的思维, 同时将二进制乘以二进制。为了 ., 我们需要将第一个二进制中的每个术语分配到第二个二进制中。

FOIL Method
::东帝汶方法

To multiply two binomials, we use the FOIL method.
::要乘以两个二进制, 我们使用FOIL方法。

$\begin{aligned} (a + b) (c + d) & = a c First, the first term in each binomial \\ + a d Outer, the outside terms in this multiplication \\ + b c Inner, the inside terms in this multiplication \\ + b d Last, the last term in each binomial \\ (a + b) (c + d) = a c + a d + b c + b d \end{aligned}$

:a+b) (c+d) = acFirst, 每一个二进制+外向器的第一个学期, 这个乘数+bcInner 的外部术语, 这个乘数+bdLast 的内在术语, 每一个二进制( a+b(c+d) = a+ad+bc+bd) = a+bc+bd

Example 4
::例4

Find $\begin{aligned} (2 x + 5) (x - 7) \end{aligned}$ .
::查找(2x+5)(x-7)

Solution: We use the FOIL method to organize our distributing of the terms in the first binomial to the second binomial and then add up the terms.
::解决方案:我们使用FOIL方法, 将术语在第一个二进制词和第二个二进制词中进行分配, 然后将术语加起来。

$\begin{aligned} F: 2 x \cdot x = 2 x^{2} \\ O: 2 x \cdot - 7 = - 14 x \\ I: 5 \cdot x = 5 x \\ L: 5 \cdot - 7 = - 35 \\ (2 x + 5) (x - 7) & = 2 x^{2} - 14 x + 5 x - 35 = 2 x^{2} - 9 x - 35 \end{aligned}$

::F: 2xxxx=2x2x2O: 2x=7}14xI: 5xx=5xL: 5*7*35(2x+5)(x-7)=2x2-14x+5x-35=2x2-9x-35

by CK-12 demonstrates how to multiply binomials .
::由 CK-12 演示如何乘以二元论。

Example 5
::例5

Find $\begin{aligned} (7 - 3 x)^{2} \end{aligned}$ .
::查找( 7- 3x) 2。

Solution: Since $\begin{aligned} (7 - 3 x)^{2} = (7 - 3 x) (7 - 3 x) \end{aligned}$ , we can use the FOIL method to square this binomial.
::解答:既然(7-3x)2=(7-3x)(7-3x)(7-3x),我们可以使用FOIL方法来平准这个二进制。

$\begin{aligned} F: 7 \cdot 7 = 49 \\ O: 7 \cdot - 3 x = - 21 x \\ I: - 3 x \cdot 7 = - 21 x \\ L: - 3 x \cdot - 3 x = 9 x^{2} \\ (7 - 3 x) (7 - 3 x) & = 49 - 21 x - 21 x + 9 x^{2} = 49 - 42 x + 9 x^{2} \end{aligned}$

::F: 7_7=49O: 73x=49O: 73x_21xI: -3x7=21xL: - 3x_3x3x=9x2x7-3x(7-3x)=49-21x-21x+9x2=49-42x9x2

Example 6
::例6

You are asked to frame a picture. You want the width of the frame to be 5 inches longer than the width of the glass and the height of the frame to be 7 inches longer than the height of the glass. You measure the glass and find the height to width ratio is 4:3. Describe the area that the picture in the frame will cover.
::您被要求设置图片框。您希望框架的宽度比玻璃宽度长5英寸,框架的高度比玻璃的高度长7英寸。您可以测量玻璃, 并发现其高度与宽度之比是 4: 3 。请描述框中图片覆盖的区域。

Solution: Since we do not know the exact dimensions of the picture, we can use x to represent the scale of the picture. That means the height will be $\begin{aligned} 4 x \end{aligned}$ and the width is $\begin{aligned} 3 x \end{aligned}$ . So, the frame's height is $\begin{aligned} 4 x + 7 \end{aligned}$ and the frame's width is $\begin{aligned} 3 x + 5 \end{aligned}$ . The area is the height times the width.
::解答: 由于我们不知道图片的准确尺寸, 我们可以使用 x 来表示图片的大小。这意味着高度为 4x, 宽度为 3x。因此, 框架的高度为 4x+7, 框架的宽度为 3x+5 。区域是宽度的高度倍数。

$\begin{aligned} A & = (4 x + 7) (3 x + 5) \\ = (4 x \cdot 3 x) + (4 x \cdot 5) + (7 \cdot 3 x) + (7 \cdot 5) \\ = 12 x^{2} + 20 x + 21 x + 35 \\ = 12 x^{2} + 41 x + 35 \end{aligned}$

::A=( 4x+7)( 3x+5) =( 4x3x) +( 4x) +( 4x) 5) +( 7x3x) +( 7}5) +( 7= 12x2+20x+21x+35= 12x2+41x+35

by Mathispower4u demonstrates how to determine the area of the shaded region by using our multiplication and subtraction of polynomials skills.
::Mathispower4u 展示了如何利用我们多面体技术的乘法和减法来确定阴影地区的面积。

Summary
::摘要

To multiply a monomial by a polynomial distribute the monomial to each one of the terms in the polynomial and use the product rule of exponents.
::将单体体乘以多面体将单体体体分配到多面体中每个术语中,并使用推手的产品规则。

To multiply a binomial by a binomial use the FOIL method to organize distributing the terms in the first binomial to the second binomial.
::将二进制乘以二进制使用FOIL方法组织将第一个二进制的术语分配给第二个二进制的术语

Review
::回顾

Multiply the polynomial expressions.
::乘以多边表达式。

1. $\begin{aligned} 5 x (x^{2} - 6 x + 8) \end{aligned}$
::1. 5x(x2-6x+8)

2. $\begin{aligned} - x^{2} (8 x^{3} - 11 x + 20) \end{aligned}$
::2.-x2(8x3-11x+20)

3. $\begin{aligned} 7 x^{3} (3 x^{3} - x^{2} + 16 x + 10) \end{aligned}$
::3. 7x3(3x3-x2+16x+10)

4. $\begin{aligned} (x^{2} + 4) (x - 5) \end{aligned}$
::4. (x2+4)(x-5)

5. $\begin{aligned} (3 x^{2} - 4) (2 x - 7) \end{aligned}$
::5. (3x2-4)(2x-7)

6. $\begin{aligned} (9 - x^{2}) (x + 2) \end{aligned}$
::6. (9-x2(x+2))

7. $\begin{aligned} (5 x - 12)^{2} \end{aligned}$
::7. (5x-12)2

8. $\begin{aligned} (4 x + 9)^{2} \end{aligned}$
::8. (4x+9)2

9. $\begin{aligned} - x^{4} (2 x + 11) (3 x^{2} - 1) \end{aligned}$
::9.-x4(2xx+11)(3x2-1)

10. $\begin{aligned} - \frac{1}{2} x^{3} (4 x^{2} + 6 x) (- 9 x + 5) \end{aligned}$
::10.-12x3(4x2+6x)(-9x+5)

Explore More
::探索更多

1. T he formula for the volume of a pyramid is $\begin{aligned} V = \frac{1}{3} B h \end{aligned}$ , where B is the area of the base of the pyramid and h is the pyramid's height. What if the area of the base of a pyramid were $\begin{aligned} x^{2} + 6 x + 8 \end{aligned}$   and the height were $\begin{aligned} 9 x \end{aligned}$ ? What would the volume of the pyramid be?
::1. 金字塔体积的公式是V=13Bh,B是金字塔底部的面积,h是金字塔的高度;如果金字塔底部的面积是x2+6x+8,而高度是9x,金字塔的体积是多少?

2. Find the volume of the figure below.
::2. 下图的数值如下。

3. Suppose a factory needs to increase the number of units it outputs. Currently it has w workers, and on average, each worker outputs u units. If it increases the number of workers by 100 and makes changes to its processes so that each worker outputs 20 more units on average, how many total units will it output?
::3. 假设一个工厂需要增加其产出单位的数量,目前它有工人,平均每个工人产出单位,如果它将工人人数增加100人并改变其流程,使每个工人平均多生产20个单位,它将产出多少个单位?

Answers to Review and Explore More Problems
::对审查和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

章节大纲

Multiplying a Polynomial by a Monomial ::以单声道乘以多声道

Example 1 ::例1

WARNING ::警告

Example 2 ::例2

Example 3 ::例3

WARNING ::警告

FOIL Method ::东帝汶方法

FOIL Method ::东帝汶方法

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers to Review and Explore More Problems ::对审查和探讨更多问题的答复

Multiplying a Polynomial by a Monomial
::以单声道乘以多声道

Example 1
::例1

WARNING
::警告

Example 2
::例2

Example 3
::例3

WARNING
::警告

FOIL Method
::东帝汶方法

FOIL Method
::东帝汶方法

Example 4
::例4

Example 5
::例5

Example 6
::例6

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers to Review and Explore More Problems
::对审查和探讨更多问题的答复