9类RD Sharma解决方案–第六章多项式的因式分解-练习6.1

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)3x ² – 4x + 15

(ii)y ² +2√3

(iii)3√x+√2x

(iv)x – 4 / x

(v)x ¹² + y ³ + t ⁵⁰

解决方案：

(i) 3x² – 4x + 15

It is a polynomial in one variable, that is, x. And all the powers of x are whole numbers.

(ii) y² + 2√3

It is a polynomial in variable y. And all the powers of y are whole numbers.

(iii) 3√x + √2x

It is not a polynomial since the exponent of 3√x is a rational term.

(iv) x – 4/x

It is not a polynomial since the exponent of variable x is – 4/x which is not a positive term.

(v) x¹² + y³ + t⁵⁰

It is a three variable polynomial, where the variables are x, y and t.

问题2：将x ²的系数写在以下每一个中：

(i)17 – 2x + 7x ²

(ii)9 – 12x + x ³

(iii) π / 6 x ² – 3x + 4

(iv)√3x– 7

解决方案：

(i) 17 – 2x + 7x²

Coefficient of x² in the above equation = 7

(ii) 9 – 12x + x³

Coefficient of x² =0, since there is no term with x²

(iii) π/6 x² – 3x + 4

Coefficient of x2 in the above equation = π/6

(iv) √3x – 7

Coefficient of x2 = 0, since there is no term with x2 in the above equation.

问题3：写下以下每个多项式的次数：

(i)7x ³ + 4x ² – 3x + 12

(ii)12 – x + 2x ³

(iii)5年–√2

(iv)7

(v)0

解决方案：

The degree is the highest possible degree of the variable in the polynomial. Now, we have

(i) Degree of the polynomial 7x³ + 4x² – 3x + 12 is 3, since the term x³ is highest

(ii) Degree of the polynomial 12 – x + 2x³ is 3, since the term x³ is highest

(iii) Degree of the polynomial 5y – √2 is 1, since the term 5y only has the variable.

(iv) Degree of the polynomial 7 is 0, since there is no term with variable.

(v) Degree of the polynomial 0 is undefined.

问题4：将以下多项式分类为线性，二次，三次和双二次多项式：

(i)x + x ² + 4

(ii)3x – 2

(iii)2x + x ²

(iv)3年

(v)t ² +1

(vi)7t ⁴ + 4t ³ + 3t – 2

解决方案：

Linear polynomials have highest degree = 1. Quadratic have have highest degree = 2. Cubic polynomials have have highest degree = 3 and bi-quadratic as 4.

(i) x + x² + 4: It is a quadratic polynomial since its highest possible degree is 2.

(ii) 3x – 2 : It is a linear polynomial since its highest possible degree is 1.

(iii) 2x + x²: It is a quadratic polynomial since its highest possible degree is 2.

(iv) 3y: It is a linear polynomial since its highest possible degree is 1.

(v) t²+ 1: It is a quadratic polynomial since its highest possible degree s 2.

(vi) 7t⁴ + 4t³ + 3t – 2: It is a bi-quadratic polynomial since its highest possible degree is 4.