问题1：写下以下每个多项式的次数：
(i)2x 3 + 5x 2 – 7
(ii)5x 2 – 3x + 2
(iii)2x + x 2 – 8
(iv)1 / 2y 7 – 12y 6 + 48y 5 – 10
(v)3x 3 +1
(vi)5
(vii)20x 3 + 12x 2 y 2 – 10y 2 + 20
解决方案：
In a polynomial, degree is the highest power of the variable.
(i) 2x3 + 5x2 – 7
Given: 2x3 + 5x2 – 7
Therefore, the degree of the polynomial, 2x3 + 5x2 – 7 is 3.
(ii) 5x2 – 3x + 2
Given: 5x2 – 3x + 2
Therefore, the degree of the polynomial, 5x2 – 3x + 2 is 2.
(iii) 2x + x2 – 8
Given: 2x + x2 – 8
Therefore, the degree of the polynomial, 2x + x2 – 8 is 2.
(iv) 1/2y7 – 12y6 + 48y5 – 10
Given: 1/2y7 – 12y6 + 48y5 – 10
Therefore, the degree of the polynomial, 1/2y7 – 12y6 + 48y5 – 10 is 7.
(v) 3x3 + 1
Given: 3x3 + 1
Therefore, the degree of the polynomial, 3x3 + 1 is 3
(vi) 5
Given: 5
Therefore, the degree of the polynomial, 5 is 0 as 5 is a constant number.
(vii) 20x3 + 12x2y2 – 10y2 + 20
Given: 20x3 + 12x2y2 – 10y2 + 20
Therefore, the degree of the polynomial, 20x3 + 12x2y2 – 10y2 + 20 is 4.
问题2：以下哪个表达式不是多项式?
(i)x 2 + 2x -2
(ii)√(ax)+ x 2 – x 3
(iii) 3年3 –√5年+ 9
(iv)斧头1/2 +斧头+ 9x 2 + 4
(v)3x -2 + 2x -1 + 4x + 5
解决方案：
(i) x2 + 2x-2
Given: x2 + 2x-2
Since variable x has a power of -2 which is negative and as a polynomial does not contain any negative powers or fractions.
Therefore, the given expression is not a polynomial.
(ii) √(ax) + x2 – x3
Given: √(ax) + x2 – x3
Since variable x has a power of 1/2 which is a fraction and as a polynomial does not contain any negative powers or fractions.
Therefore, the given expression is not a polynomial.
(iii) 3y3 – √5 y + 9
Given: 3y3 – √5 y + 9
Since the polynomial has positive powers i.e. non-negative integers.
Therefore, the given expression is a polynomial.
(iv) ax1/2 + ax + 9x2 + 4
Given: ax1/2 + ax + 9x2 + 4
Since variable x has a power of 1/2 which is a fraction and as a polynomial does not contain any negative powers or fractions.
Therefore, the given expression is not a polynomial.
(v) 3x-2 + 2x-1 + 4x + 5
Given: 3x-2 + 2x-1 + 4x + 5
Since variable x has a power of -2 and -1 which are negative and as a polynomial does not contain any negative powers or fractions.
The given expression is not a polynomial.
问题3：将以下每个多项式写在标准中。另外，写出他们的学位：
(i)x 2 + 3 + 6x + 5x 4
(ii) 2 + 4 + 5a 6
(ⅲ)(X 3 – 1)(X 3 – 4)
(iv)(y 3 – 2)(y 3 + 11)
(v)(a 3 – 3/8)(a 3 + 16/17)
(六)(一+ 3/4)(一+ 4/3)
解决方案：
(i) x2 + 3 + 6x + 5x4
Given: x2 + 3 + 6x + 5x4
Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.
Therefore, the expressions are:
(3 + 6x + x2 + 5x4) or (5x4 + x2 + 6x + 3)
The degree of the given polynomial is 4.
(ii) a2 + 4 + 5a6
Given: a2 + 4 + 5a6
Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.
Therefore, the expressions are:
(4 + a2 + 5a6) or (5a6 + a2 + 4)
The degree of the given polynomial is 6.
(iii) (x3 – 1) (x3 – 4)
Given: (x3 – 1) (x3 – 4)
x6 – 4x3 – x3 + 4
x6 – 5x3 + 4
Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.
Therefore, the expressions are:
(4 – 5x3 + x6) or (x6 – 5x3 + 4)
The degree of the given polynomial is 6.
(iv) (y3 – 2) (y3 + 11)
Given: (y3 – 2) (y3 + 11)
y6 + 11y3 – 2y3 – 22
y6 + 9y3 – 22
Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.
Therefore, the expressions are:
(-22 + 9y3 + y6) or (y6 + 9y3 – 22)
The degree of the given polynomial is 6.
(v) (a3 – 3/8) (a3 + 16/17)
Given: (a3 – 3/8) (a3 + 16/17)
a6 + 16a3/17 – 3a3/8 – 6/17
a6 + (77/136)a3 – 48/136
Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.
Therefore, the expressions are:
(-48/136 + (77/136)a3 + a6) or (a6 + (77/136)a3 – 48/136)
The degree of the given polynomial is 6.
(vi) (a + 3/4) (a + 4/3)
Given: (a + 3/4) (a + 4/3)
a2 + 4a/3 + 3a/4 + 1
a2 + (25/12)a + 1
Since the standard form of the polynomial can be written in either increasing or decreasing order of their powers.
Therefore, the expressions are:
(1 + (25/12)a + a2) or (a2 + (25/12)a + 1)
The degree of the given polynomial is 2.