Section: 添加和减减多面体 | 6.4 添加和减减多面体

Section outline

Rectangular prism A has a volume of $\begin{aligned} x^{3} + 2 x^{2} - 3 \end{aligned}$ . Rectangular prism B has a volume of $\begin{aligned} x^{4} + 2 x^{3} - 8 x^{2} \end{aligned}$ . How much larger is the volume of rectangular prism B than rectangular prism A?
::矩形棱柱 A 的体积为 x3+2x2-3。矩形棱柱 B 的体积为 x4+2x3-8x2。矩形棱柱 B 的体积比矩形棱柱 A 的体积大多少?

Adding and Subtracting Polynomials
::添加和减减多面体

A polynomial is an expression with multiple variable terms , such that the exponents are greater than or equal to zero. All quadratic and linear equations are . Equations with negative exponents, square roots, or variables in the denominator are not polynomials.
::多面性是一个表达式,有多个变量, 表示符大于或等于零。所有二次方程和线性方程都是。负引号、平根或分母变量的等式不是多面性。

Now that we have established what a polynomial is, there are a few important parts. Just like with a quadratic, a polynomial can have a constant , which is a number without a variable. The degree of a polynomial is the largest exponent . For example, all quadratic equations have a degree of 2. Lastly, the leading coefficient is the coefficient in front of the variable with the degree. In the polynomial $\begin{aligned} 4 x^{4} + 5 x^{3} - 8 x^{2} + 12 x + 24 \end{aligned}$ above, the degree is 4 and the leading coefficient is also 4. Make sure that when finding the degree and leading coefficient you have the polynomial in standard form . Standard form lists all the variables in order , from greatest to least.
::现在,我们已经确定了多面体是什么, 有几个重要部分。就像四面体一样, 多面体可以有一个常数, 这是一个没有变量的数字。多面体的程度是最大的引号。例如, 所有四面体方程式的度为 2 。最后, 主要的系数是变量前的系数。在以上多面体 4x4+5x3- 8x2+12x+24 中, 度为 4, 主要系数也是 4 。当找到学位和主要系数时, 请确保您有标准形式的多面体。标准格式列出了所有变量, 从最大到最小的变量。

Let's rewrite $\begin{aligned} x^{3} - 5 x^{2} + 12 x^{4} + 15 - 8 x \end{aligned}$ in standard form and find the degree and leading coefficient.
::让我们重写标准格式的 x3- 5x2+12x4+15-8x, 找到程度和主要系数。

To rewrite in standard form, put each term in order, from greatest to least, according to the exponent. Always write the constant last.
::以标准格式重写, 将每个词按顺序排列, 从最大到最小, 根据表情。总是写最后一个常数。

$\begin{aligned} x^{3} - 5 x^{2} + 12 x^{4} + 15 - 8 x \to 12 x^{4} + x^{3} - 5 x^{2} - 8 x + 15 \end{aligned}$

::x3-5x2+12x4+15-8x12x4+x3-5x2-8x15

Now, it is easy to see the leading coefficient, 12, and the degree, 4.
::现在,很容易看到主要系数12和程度4。

Simplify $\begin{aligned} (4 x^{3} - 2 x^{2} + 4 x + 15) + (x^{4} - 8 x^{3} - 9 x - 6) \end{aligned}$
::简化 (4x3 - 2x2+4x+15)+( x4 - 8x3 - 9x-6)

To add or subtract two polynomials, combine like terms . Like terms are any terms where the exponents of the variable are the same. We will regroup the polynomial to show the like terms.
::要添加或减去两个多义, 请将类似术语合并。类似术语是变量的出处相同的任何术语。我们将重新组合多义以显示类似术语。

$\begin{aligned} (4 x^{3} - 2 x^{2} + 4 x + 15) + (x^{4} - 8 x^{3} - 9 x - 6) \\ x^{4} + (4 x^{3} - 8 x^{3}) - 2 x^{2} + (4 x - 9 x) + (15 - 6) \\ x^{4} - 4 x^{3} - 2 x^{2} - 5 x + 9 \end{aligned}$

:4x3-2x2+4x+15+15)+(x4-8x3-8x3-9x-6)x4+(4x3-8x3-8x3)-2x2+(4x-9x)+(15-6)x4-4x3-2x2-5x+9)

Simplify $\begin{aligned} (2 x^{3} + x^{2} - 6 x - 7) - (5 x^{3} - 3 x^{2} + 10 x - 12) \end{aligned}$
::简化 (2x3+x2- 6x- 7)- (5x3- 3x2+10x- 12)

When subtracting, distribute the negative sign to every term in the second polynomial, then combine like terms.
::减法时, 将负符号分布到第二个多义词中的每个词, 然后将类似词合并。

$\begin{aligned} (2 x^{3} + x^{2} - 6 x - 7) - (5 x^{3} - 3 x^{2} + 10 x - 12) \\ 2 x^{3} + x^{2} - 6 x - 7 - 5 x^{3} + 3 x^{2} - 10 x + 12 \\ (2 x^{3} - 5 x^{3}) + (x^{2} + 3 x^{2}) + (- 6 x - 10 x) + (- 7 + 12) \\ - 3 x^{3} + 4 x^{2} - 16 x + 5 \end{aligned}$

:2x3+x2-6x-2-6x-7)-(5x3-3x2-3x2+10x2-12)-2x3+6x-7-5x3+3x2-10x12(2x3-5x3)+(x2+3x3-3x3)+(6x-3x2)+(-6x-10x)+(-7+12)-3x3+3x3+4x2-16x5)

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the difference in the volume of rectangular prism B compared to rectangular prism A.
::早些时候,有人要求你找到长方形棱柱B与长方形棱柱A的体积差异。

We need to subtract the volume of rectangular prism A from the volume of rectangular prism B.
::我们需要从矩形棱柱B的体积中减去矩形棱柱A的体积。

$\begin{aligned} (x^{4} + 2 x^{3} - 8 x^{2}) - (x^{3} + 2 x^{2} - 3) \\ = x^{4} + 2 x^{3} - 8 x^{2} - x^{3} - 2 x^{2} + 3 \\ = x^{4} + x^{3} - 10 x^{2} + 3 \end{aligned}$

:x4+2x3-8x2)-(x3+2x2-2-3)=x4+2x3-8x3-8x2-x3--2x2+3=x4+x3-10x2+3)

Therefore , the difference between the two volumes is $\begin{aligned} x^{4} + x^{3} - 10 x^{2} + 3 \end{aligned}$ .
::因此,这两卷之间的差额是x4+x3-10x2+3。

Example 2
::例2

Is $\begin{aligned} \sqrt{2 x^{3} - 5 x} + 6 \end{aligned}$ a polynomial? Why or why not?
::2x3- 5x+6 是多面体吗? 为什么或为什么不是?

No, this is not a polynomial because $\begin{aligned} x \end{aligned}$ is under a square root in the equation .
::不,这不是一个多数值,因为 x 在方程式的平方根下。

Example 3
::例3

Find the leading coefficient and degree of $\begin{aligned} 6 x^{2} - 3 x^{5} + 16 x^{4} + 10 x - 24 \end{aligned}$ .
::查找6x2-3x5+16x4+10x-24的主要系数和程度。

In standard form, this polynomial is $\begin{aligned} - 3 x^{5} + 16 x^{4} + 6 x^{2} + 10 x - 24 \end{aligned}$ . Therefore, the degree is 5 and the leading coefficient is -3.
::在标准格式中,该多数值为-3x5+16x4+6x2+10x-24,因此,学位为5,主要系数为-3。

Example 4
::例4

Add the following polynomials: $\begin{aligned} (9 x^{2} + 4 x^{3} - 15 x + 22) + (6 x^{3} - 4 x^{2} + 8 x - 14) \end{aligned}$ .
::增加以下多数值9x2+4x3-15x+22)+(6x3-4x2+8x-14)。

$\begin{aligned} (9 x^{2} + 4 x^{3} - 15 x + 22) + (6 x^{3} - 4 x^{2} + 8 x - 14) = 10 x^{3} + 5 x^{2} - 7 x + 8 \end{aligned}$
:9x2+4x3-15x+22)+(6x3-4x2+8x-14)=10x3+5x2-7x+8)

Example 5
::例5

Subtract the following polynomials: $\begin{aligned} (7 x^{3} + 20 x - 3) - (x^{3} - 2 x^{2} + 14 x - 18) \end{aligned}$ .
::减去下列多数值7x3+20x-3)-(x3-2x2+14x-18)。

$\begin{aligned} (7 x^{3} + 20 x - 3) - (x^{3} - 2 x^{2} + 14 x - 18) = 6 x^{3} + 2 x^{2} + 6 x + 15 \end{aligned}$
:7x3+20x-3)-(x3-2x2+14x-18)=6x3+2x2+6x+15

Review
::回顾

Determine if the following expressions are polynomials. If not, state why. If so, write in standard form and find the degree and leading coefficient.
::确定以下表达式是否为多数值。如果不是, 请说明原因。如果不是, 请以标准格式写入, 并找到程度和主要系数。

$\begin{aligned} \frac{1}{x^{2}} + x + 5 \end{aligned}$
::1x2+x+5

$\begin{aligned} x^{3} + 8 x^{4} - 15 x + 14 x^{2} - 20 \end{aligned}$
::x3+8x4 - 15x+14x2 - 20

$\begin{aligned} x^{3} + 8 \end{aligned}$
::x3+8x3+8

$\begin{aligned} 5 x^{- 2} + 9 x^{- 1} + 16 \end{aligned}$
::5x-2+9x-1+16

$\begin{aligned} x^{2} \sqrt{2} - x \sqrt{6} + 10 \end{aligned}$
::x22 - x6+10 x22 - x6+10

$\begin{aligned} \frac{x^{4} + 8 x^{2} + 12}{3} \end{aligned}$
::x4+8x2+123

$\begin{aligned} \frac{x^{2} - 4}{x} \end{aligned}$
::x2 - 4x

$\begin{aligned} - 6 x^{3} + 7 x^{5} - 10 x^{6} + 19 x^{2} - 3 x + 41 \end{aligned}$
::- 6x3+7x5-10x6+19x2-3x+41

Add or subtract the following polynomials.
::添加或减去下列多数值。

$\begin{aligned} (x^{3} + 8 x^{2} - 15 x + 11) + (3 x^{3} - 5 x^{2} - 4 x + 9) \end{aligned}$
:x3+8x2 - 15x+11)+(3x3 - 5x2 - 4x+9)

$\begin{aligned} (- 2 x^{4} + x^{3} + 12 x^{2} + 6 x - 18) - (4 x^{4} - 7 x^{3} + 14 x^{2} + 18 x - 25) \end{aligned}$
:-2x4+x3+12x2+6x-18)-(4x4-7x3+14x2+18x-25)

$\begin{aligned} (10 x^{3} - x^{2} + 6 x + 3) + (x^{4} - 3 x^{3} + 8 x^{2} - 9 x + 16) \end{aligned}$
:10x3-x2+6x+3)+(x4-3x3+8x2-9x+16)

$\begin{aligned} (7 x^{3} - 2 x^{2} + 4 x - 5) - (6 x^{4} + 10 x^{3} + x^{2} + 4 x - 1) \end{aligned}$
:7x3-2x2+4x-5)-(6x4+10x3+x2+4x-1)

$\begin{aligned} (15 x^{2} + x - 27) + (3 x^{3} - 12 x + 16) \end{aligned}$
:15x2+x-27)+(3x3-12x+16)

$\begin{aligned} (2 x^{5} - 3 x^{4} + 21 x^{2} + 11 x - 32) - (x^{4} - 3 x^{3} - 9 x^{2} + 14 x - 15) \end{aligned}$
:2x5-3x4+21x2+11x-32)-(x4-3x3-9x2+14x-15)

$\begin{aligned} (8 x^{3} - 13 x^{2} + 24) - (x^{3} + 4 x^{2} - 2 x + 17) + (5 x^{2} + 18 x - 19) \end{aligned}$
:8x3-13x2+24)-(x3+4x2-2x+17)+(5x2+18x-19)

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

Adding and Subtracting Polynomials ::添加和减减多面体

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Adding and Subtracting Polynomials
::添加和减减多面体

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)