5.2 Polynomials

​Polynomials 多項式

​

Multiple Select

5a is a ...?

5a 是一個……？

• algebraic

  代數式

• monomial

  單項式

• equations

  等式、方程

• formula

  公式

​Monomials 單項式

​a kind of algebraic expression 一種代數式

​

• a number 一個數字

• the product of a number​ and one or two variables with positive integral indices

• 數字和變數的乘積（並且變數的指數為正整數）

• if it's a fraction, no variables in denominator

• 如果是分數，分母中不能含有變數。

Multiple Select

which of the following expression is/are a monomial(s)?

以下哪個/哪些是單項式？

• 2pq

• $$\frac{x^3}{2}$$  

• $$\frac{3}{x^2}$$  

• $$-m^2n^3$$  

• $$4a^5b^x$$  

​Coeficient of the monomials

單項式的係數

​

• the numerical part (including the sign) is call the coefficient of the monomial.

  對於含有變數的單項式，變數​前的數字叫做多項式的係數。

• ​the coefficient of 3x is 3.

• the coefficient of -3x is -3.​

Draw

Write down the coefficient of 9y.

寫下單項式 9y 的係數。

Draw

Circle the coefficient.

圈住係數。

Draw

Circle the coefficient:

Draw

Write down the coeficient:

​Polynomial 多項式

a monomial, or a sum of two or more monomial​s.

單項式，或者是多個單項式相加之和。​

Draw

Write down a polynomial randomly.

嘗試隨意寫下一個多項式。

​Each monomial is called a term in a polynomial.

多項式中的每一個單項式，叫做“項”。

​

​

​If a term does not contain any variables, it is called a constant term.

不含變數的項，叫做“常數項”​。

Multiple Choice

the number of terms of 1-9x+8x²  is 3.

1-9x+8x²  含有3個項。

• True

• False

Draw

Write down the terms of the polynomial: 1-9x+8x²    

寫出多項式 1-9x+8x²  含有的項。

Fill in the Blanks

The number of terms of the following polynomials is ...?

以下多項式含有幾項？

x³+6x² 

Fill in the Blanks

The number of terms of the following polynomials is ...?

以下多項式含有幾項？

$-5-x^3+\frac{7}{4}x$  

Fill in the Blanks

The number of terms of the following polynomials is ...?

以下多項式含有幾項？

3x³+7x-4x²-2 

​Specially, 特別地，

polynomials consisting of two terms is called binomials.

含有兩項的多項式叫做“二項式”​

​

polynomials consisting of three terms is called trinomials.​

含有三項的多項式叫做“三項式”​

Draw

Create a binomial:

隨意創作一個二項式：

Draw

Create a trinomial:

隨意創作一個三項式：

​Degree 次數

• different from "power" 和次方/指數不同

• ​related with "power" 但和次方/指數相關^^​

​Degree of a monomial is the sum of the indices of all variables in the monomials.

​單項式的次數等於單項式中所有變數的指數之和

​e.g. degree of 3x is 1.

degree of 3xy is 2.​

Fill in the Blanks

Write down the degree of the following monomials:

寫下以下單項式的次數：

10x² 

Fill in the Blanks

Write down the degree of the following monomials:

寫下以下單項式的次數：

2y⁷ 

Fill in the Blanks

Write down the degree of the following monomials:

寫下以下單項式的次數：

8xy

Fill in the Blanks

Write down the degree of the following monomials:

寫下以下單項式的次數：

20x³y⁴

​The degree of a polynomial is the highest degree of all terms in the polynomial.

多項式的次數為該多項式中所有項的最高次數。​

​e.g. consider 3xy⁴-6x³y²-7x²y+4x²

​the degrees of the four terms are 5, 5, 3, 2 respectively,

多項式中四個項的次數分別是5,5,3,2。​

the highest degree is 5,

其中最高次數是5，​

so the degree of the polynomial is 5.​

因此，這個多項式的次數就是5.​

Fill in the Blanks

Write down the degree of the following polynomials:

寫出以下多項式的次數：

6x⁴-2x³

Fill in the Blanks

Write down the degree of the following polynomials:

寫出以下多項式的次數：

2x-4x²+3x⁶

Fill in the Blanks

Write down the degree of the following polynomials:

寫出以下多項式的次數：

6x³y³-xy⁴+12

Fill in the Blanks

Write down the degree of the following polynomials:

寫出以下多項式的次數：

16x-20x⁴y+20xy²

​Ascending power 升冪:

​x的次方/指數越來越大 e.g. 2+5x+x²-x³+3x⁴

&

Descending power 降冪：

​x的次方/指數越來越小 e.g. 3x⁴-x³+x²+5x+2

​Polynomial in one variable 一元多項式：

只含有一個變數的多項式​

e.g. 2+5x+x²-x³+3x⁴​

Multiple Choice

Determine the terms of the polunomial is in ascending or descending power of x:

判斷以下多項式是以x的升冪還是降冪排列：

a⁵-4a⁴+10a²+7a

• ascending power

• descending power

Multiple Choice

Determine the terms of the polunomial is in ascending or descending power of x:

判斷以下多項式是以x的升冪還是降冪排列：

2x+7x²-6x⁴

• ascending power

• descending power

Draw

Arrange the terms of the polynomial in ascending power of x

將下列多項式按 x 的升冪排列：

3x-1+5x³

Draw

Arrange the terms of the polynomial in descending power of x

將下列多項式按 x 的降冪排列：

-2x+3-5x²

​Value of a polynomial: 多項式的值

即：多項式等於多少？​

​We can use: method of substitution 代入法

Draw

Find the value of the polynomial: 4x+1 when x=5.

當x=5時，求多項式4x+1的值。

​Class Summary 課堂小總結

What we've learned:

• Monomials 單項式

• Polynomials 多項式

• Coefficient of monomials 單項式的係數

• degree of monomials and polynomials 單項式和多項式的次數

• arrangement of ascending power and descending power of polynomials 升冪/降冪排列

• use method of substitution to find out the value of polynomials 代入法找多項式的值

Poll

I can understand the following topic through out the lesson:

通過這節課，你明白了……

• What Monomials and Polynomials are

• How to find out the Coefficient of a monomial

• How to find out the degree of a monomial or a polynomial

• How to arrange a polynomial in ascending power and descending power

• Use method of substitution to find out the value of a monomial or a polynomial