9类NCERT解决方案–第2章多项式–练习2.1

问题1.以下哪个表达式是一个变量的多项式，而哪些不是?请说明您回答的原因。

(i)4x ² – 3x + 7
(ii)y ² +√2
(iii)3√t+t√2
(iv)y + 2 / y
(v)x ¹⁰ + y ³ + t ⁵⁰

解决方案：

(i) The algebraic expression 4x² – 3x + 7 can be written as 4x² – 3x + 7x⁰

As we can see, all exponents of x are whole numbers, 

So, the given expression 4x² – 3x + 7 is polynomial in one variable.

(ii) The algebraic expression y² + √2 can be written as y² + √2y⁰

As we can see, all exponents of y are whole numbers, 

So, the given expression y² + √2 is polynomial in one variable.

(iii) The algebraic expression 3 √t + t√2 can be written as 3 t^1/2 + √2.t

As we can see, one exponent of t is 1/2, which is not a whole number,

So, the given expression 3 √t + t√2 is not a polynomial in one variable.

(iv) The algebraic expression y + 2/y can be written as y + 2.y^-1

As we can see, one exponent of y is -1, which is not a whole number,

So, the given expression y+ 2/y is not a polynomial in one variable.

(v) The given algebraic expression is x¹⁰+ y³+ t⁵⁰

As we can see, the expression contains three variables i.e x, y, and t,

So, the given expression x¹⁰ + y³ + t⁵⁰ is not a polynomial in one variable.

问题2.在下面的每一个中写x ^2的系数

(i)2 + x ² + x
(ii)2 – x ² + x ³
(iii)pi / 2 x ² + x
(iv)√2x– 1

解决方案：

(i) The given algebraic expression is 2 + x² + x

As we can clearly see, the coefficient of x² is 1.

(ii) The given algebraic expression is 2 – x² + x³

As we can clearly see, the coefficient of x² is -1.

(iii) The given algebraic expression is pi/2 x² + x

As we can clearly see, the coefficient of x² is pi/2.

(iv) The given algebraic expression is √2 x — 1

As we can clearly see, the coefficient of x² is 0.

问题3。举一个例子，一个二项式的次数为35，一个二项式的次数为100。

解决方案：

A Binomial having degree 35 is 4x³⁵ + 50

A Monomial having degree 100 is 3t¹⁰⁰pi

问题4.写下以下每个多项式的次数

(i)5倍³ + 4倍² + 7倍
(ii)4 – y ²
(iii)5吨–√7
(iv)3

解决方案：

The highest power of a variable in the given expression is known as the Degree of the polynomial 

(i) The given expression is 5x³ + 4x² + 7x

As we can clearly see, the highest power of variable x is 3,

So, the degree of given polynomial 5x³+4x² + 7x is 3.

(ii) The given expression is 4 – y²

As we can clearly see, the highest power of variable y is 2,

So, the degree of given polynomial 4 – y² is 2.

(iii) The given expression is 5t – √7

As we can clearly see, the highest power of variable t is 1,

So, the degree of given polynomial 5t – √7 is 1.

(iv) The given expression 3 can be written as 3x⁰

As we can clearly see, the highest power of variable x is 0,

So, the degree of given polynomial 3 is 0.

问题5.将以下项分类为线性，二次和三次多项式

(i)x ² + x
(ii)x – x ³
(iii)y + y ² + 4
(iv)1 + x
(v)3吨
(vi)r ²
(vii)7倍³

解决方案：

(i) Since the degree of given polynomial x² + x is 2,

So, it is a Quadratic Polynomial.

(ii) Since the degree of given polynomial x – x³ is 3,

So, it is a Cubic Polynomial.

(iii) Since the degree of given polynomial y + y² + 4 is 2,

So, it is a Quadratic Polynomial.

(iv) Since the degree of given polynomial 1 + x is 1,

So, it is a Linear Polynomial.

(v) Since the degree of given polynomial 3t is 1, 

So, it is a Linear Polynomial.

(vi) Since the degree of given polynomial r² is 2,

So, it is a Quadratic Polynomial.

(vii) Since the degree of given polynomial 7x³ is 3,

So, it is a Cubic Polynomial.