Section: 单体体在多面体中的乘法 | 9.3 单体体多面体的倍增

Section outline

Multiplication of Monomials by Polynomials
::单体体在多面体中的乘法

Just as we can add and subtract , we can also multiply them. The and the techniques you’ve learned for dealing with exponents will be useful here.
::正如我们可以增减一样,我们也能够乘以它们。你所学到的对付表率的方法和技巧在这里将很有用。

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

When multiplying polynomials, remember the exponent rules , particularly the product rule: $\begin{aligned} x^{n} \cdot x^{m} = x^{n + m} \end{aligned}$ .
::当乘多数值时,请记住前列规则,特别是产品规则:xnxm=xn+m。

If the expressions have coefficients and more than one variable , multiply the coefficients just as you would any number and apply the product rule on each variable separately.
::如果表达式有系数和一个以上变量,则与任何数字一样乘以系数,并对每个变量分别适用产品规则。

Multiplying Monomials
::乘数单数

Multiply the following monomials.
::乘以下列单数。

a) $\begin{aligned} (2 x^{2}) (5 x^{3}) \end{aligned}$
:a) (2x2(5x3))

$\begin{aligned} (2 x^{2}) (5 x^{3}) = (2 \cdot 5) \cdot (x^{2} \cdot x^{3}) = 10 x^{2 + 3} = 10 x^{5} \end{aligned}$
:2x2( 5x3) = (2) 5x5) = (x2xx3) = 10x2+3= 10x5

b) $\begin{aligned} (- 3 y^{4}) (2 y^{2}) \end{aligned}$
:--3y4)(2y2)

$\begin{aligned} (- 3 y^{4}) (2 y^{2}) = (- 3 \cdot 2) \cdot (y^{4} \cdot y^{2}) = - 6 y^{4 + 2} = - 6 y^{6} \end{aligned}$
:-3y4)(2y2)=(-32) (y4y2) 6y4+26y6)

c) $\begin{aligned} (3 x y^{5}) (- 6 x^{4} y^{2}) \end{aligned}$
:c) (3xy5)(-6x4y2)

$\begin{aligned} (3 x y^{5}) (- 6 x^{4} y^{2}) = - 18 x^{1 + 4} y^{5 + 2} = - 18 x^{5} y^{7} \end{aligned}$
:3xy5 (- 6x4y2) 18x1+4y5+218x5y7)

d) $\begin{aligned} (- 12 a^{2} b^{3} c^{4}) (- 3 a^{2} b^{2}) \end{aligned}$
:d) (-12a2b3c4)(-3a2b2)

$\begin{aligned} (- 12 a^{2} b^{3} c^{4}) (- 3 a^{2} b^{2}) = 36 a^{2 + 2} b^{3 + 2} c^{4} = 36 a^{4} b^{5} c^{4} \end{aligned}$
:-12a2b3c4)(-3a2b2)=36a2+2b3+2c4=36a4b5c4)

To multiply a polynomial by a monomial , use the Distributive Property . Remember, that property says that $\begin{aligned} a (b + c) = a b + a c \end{aligned}$ .
::要将一个多数值乘以单数值, 请使用分配属性。记住, 该属性表示 a( b+c) =ab+ac 。

Using the Distributive Property
::使用分配财产

1. Multiply:
::1. 乘以:

a) $\begin{aligned} 3 (x^{2} + 3 x - 5) \end{aligned}$
:a) 3(x2+3x-5)

$\begin{aligned} 3 (x^{2} + 3 x - 5) = 3 (x^{2}) + 3 (3 x) - 3 (5) = 3 x^{2} + 9 x - 15 \end{aligned}$
::3(x2+3x-5)=3(x2)+3(3x)-3(5)=3x2+9x-15

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

$\begin{aligned} 4 x (3 x^{2} - 7) = (4 x) (3 x^{2}) + (4 x) (- 7) = 12 x^{3} - 28 x \end{aligned}$
::4x(3x2-7)=(4x(3x2)+(4x(7)=12x3-28x)

c) $\begin{aligned} - 7 y (4 y^{2} - 2 y + 1) \end{aligned}$
::c)-7y(4y2-2y+1)

$\begin{aligned} - 7 y (4 y^{2} - 2 y + 1) & = (- 7 y) (4 y^{2}) + (- 7 y) (- 2 y) + (- 7 y) (1) \\ = - 28 y^{3} + 14 y^{2} - 7 y \end{aligned}$

::-7y(4y2-2y+1)=(-7y(4y2)+(-7y)(-2y)+(-7y)(-7y)(-7y)+(-7y)(1)28y3+14y2-7yy

Notice that when you use the Distributive Property, the problem becomes a matter of just multiplying monomials by monomials and adding all the separate parts together.
::请注意,当您使用分配财产时,问题就变成了将单项财产乘以单项财产,并将所有单独的部分加在一起的问题。

2. Multiply:
::2. 乘以:

a) $\begin{aligned} 2 x^{3} (- 3 x^{4} + 2 x^{3} - 10 x^{2} + 7 x + 9) \end{aligned}$
:a) 2x3(- 3x4+2x3- 10x2+7x+9)

$\begin{aligned} 2 x^{3} (- 3 x^{4} + 2 x^{3} - 10 x^{2} + 7 x + 9) & = (2 x^{3}) (- 3 x^{4}) + (2 x^{3}) (2 x^{3}) + (2 x^{3}) (- 10 x^{2}) + (2 x^{3}) (7 x) + (2 x^{3}) (9) \\ = - 6 x^{7} + 4 x^{6} - 20 x^{5} + 14 x^{4} + 18 x^{3} \end{aligned}$

::2x3(- 3x4+2x3- 2x3- 10x2+7x+9) =( 2x3)(-3x4) +( 2x3)( 2x3) +( 2x3)( - 10x3) +( 2x3)( 7x) +(2x3)( 9) 6x7+4x6-205+14x4+18x3)

b) $\begin{aligned} - 7 a^{2} b c^{3} (5 a^{2} - 3 b^{2} - 9 c^{2}) \end{aligned}$
::b)-7a2bc3(5a2-3b2-9c2)

$\begin{aligned} - 7 a^{2} b c^{3} (5 a^{2} - 3 b^{2} - 9 c^{2}) & = (- 7 a^{2} b c^{3}) (5 a^{2}) + (- 7 a^{2} b c^{3}) (- 3 b^{2}) + (- 7 a^{2} b c^{3}) (- 9 c^{2}) \\ = - 35 a^{4} b c^{3} + 21 a^{2} b^{3} c^{3} + 63 a^{2} b c^{5} \end{aligned}$

::-7a2bc3(5a2-3b2-9c2)=(-7a2bc3(5a2)+(-7a2bc3)(-3b2)+(-7a2bc3)(-3b2)+(-7a2bc3)(-9c2)+(-3a2bc3)(-9c2)+(35a4bc3+21a2b3c3+63a2c5)

Example
::示例示例示例示例

Example 1
::例1

Multiply $\begin{aligned} - 2 a^{2} b^{4} (3 a b^{2} + 7 a^{3} b - 9 a + 3) \end{aligned}$ .
::乘以-2a2b4(3ab2+7a3b-9a+3)

Multiply the monomial by each term inside the parenthesis:
::以括号内的每个词乘以单数 :

$\begin{aligned} - 2 a^{2} b^{4} (3 a b^{2} + 7 a^{3} b - 9 a + 3) \\ = (- 2 a^{2} b^{4}) (3 a b^{2}) + (- 2 a^{2} b^{4}) (7 a^{3} b) + (- 2 a^{2} b^{4}) (- 9 a) + (- 2 a^{2} b^{4}) (3) \\ = - 6 a^{3} b^{6} - 14 a^{5} b^{5} + 18 a^{3} b^{4} - 6 a^{2} b^{4} \end{aligned}$

::-2a2b4(3ab2+7a3b-9a+3)=(-2a2b4(3ab2)+(-2a2b4)+(-2a2b4(7a3b)+(-2a2b4(9a)+(-2a2b4)(9a)+(-2a2b4)(3)*6a3b6-14a5b5+18a3b4-6a2b4)

Review
::回顾

Multiply the following monomials.
::乘以下列单数。

$\begin{aligned} (2 x) (- 7 x) \end{aligned}$
:2x)(-7x)

$\begin{aligned} (10 x) (3 x y) \end{aligned}$
:10x( 3xy))

$\begin{aligned} (4 m n) (0.5 n m^{2}) \end{aligned}$
:4mn)(0.50nm2)

$\begin{aligned} (- 5 a^{2} b) (- 12 a^{3} b^{3}) \end{aligned}$
:-5a2b)(-12a3b3)

$\begin{aligned} (3 x y^{2} z^{2}) (15 x^{2} y z^{3}) \end{aligned}$
:3xy2z2)(15x2yz3)

Multiply and simplify.
::乘数和简化。

$\begin{aligned} 17 (8 x - 10) \end{aligned}$
::17(8x-10)

$\begin{aligned} 2 x (4 x - 5) \end{aligned}$
::2x(4x-5)

$\begin{aligned} 9 x^{3} (3 x^{2} - 2 x + 7) \end{aligned}$
::9x3( 3x2 - 2x+7)

$\begin{aligned} 3 x (2 y^{2} + y - 5) \end{aligned}$
::3x( 2y2+y- 5)

$\begin{aligned} 10 q (3 q^{2} r + 5 r) \end{aligned}$
::10q(3q2r+5r)

$\begin{aligned} - 3 a^{2} b (9 a^{2} - 4 b^{2}) \end{aligned}$
::- 3a2b(9a2-4b2)

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

Multiplication of Monomials by Polynomials ::单体体在多面体中的乘法

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

Multiplying Monomials ::乘数单数

Using the Distributive Property ::使用分配财产

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Multiplication of Monomials by Polynomials
::单体体在多面体中的乘法

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

Multiplying Monomials
::乘数单数

Using the Distributive Property
::使用分配财产

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)