NCERT Solutions Chapter 2 - Polynomials (I), Class 9, Maths PDF Download

1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

(i) 4x² - 3x +  7

(ii) y²  + √2

(iii) 3√t +  t√2

(iv) y + 2/y

(v) x¹⁰ +  y³ +  t⁵⁰

Answer

(i) 4x² - 3x +  7
There is only one variable x with whole number power so this polynomial in one variable.

(ii)  y² + √2

There is only one variable y with whole number power so this polynomial in one variable.

(iii) 3√2  + t√2 

There is only one variable t but in 3√t power of t is 1/2 which is not a whole number so 3√t +  t√2 is not a polynomial.

(iv) y +  2/y

There is only one variable y  but 2/y = 2y-1 so the power is not a whole number so y + 2/y is not a polynomial. 

(v) x¹⁰  + y³ +  t⁵⁰

There are three variable x, y and t and there powers are whole number so this polynomial in three variable.

2. Write the coefficients of x² in each of the following:
 (i) 2 + x² + x
 (ii) 2 - x² +  x³

⁽ⁱⁱⁱ⁾ 

(iv) √2x - 1

Answer

(i) coefficients of x² = 1
(ii) coefficients of x² = -1
(iii) coefficients of x² = π/2
(iv) coefficients of x² = 0

3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Answer

3x³⁵ + 7 and 4x¹⁰⁰

4. Write the degree of each of the following polynomials:

(i) 5x³ + 4x² + 7x 

(ii) 4 – y² 

(iii) 5t – √7

(iv) 3

Answer

(i) 5x³ has highest power in the given polynomial which power is 3. Therefore, degree of polynomial is 3.

(ii) – y²  has highest power in the given polynomial which power is 2. Therefore, degree of polynomial is 2.

(iii) 5t has highest power in the given polynomial which power is 1. Therefore, degree of polynomial is 1.

(iv) There is no variable in the given polynomial. Therefore, degree of polynomial is 0.

5. Classify the following as linear, quadratic and cubic polynomial:

(i) x² + x
► Quadratic Polynomial

(ii) x - x³
► Cubic Polynomial

(iii) y  + y² + 4
► Quadratic Polynomial
 

(iv) 1 +  x
► Linear Polynomial
 

(v) 3t
►Linear Polynomial
 

(vi) r²

► Quadratic Polynomial

(vii) 7x³
► Cubic Polynomial

Exercise 2.2

 1. Find the value of the polynomial at 5x + 4x² +  3 at

 (i) x = 0 

(ii) x = - 1 

(iii) x = 2

Answer

2. Find p(0), p(1) and p(2) for each of the following polynomials:

(i) p(y) = y² - y +1

(ii) p(t) = 2 +t + 2t² - t³
 (iii) p(x) = x³ 

(iv) p(x) = (x - 1) (x + 1)

Answer

3. Verify whether the following are zeroes of the polynomial, indicated against them.

(i) p(x) = 3x  1, x = -1/3

(ii)  p(x) = 5x - π, x = 4/5

(iii) p(x) = x² - 1, x = 1, -1

(iv) p(x) = (x + 1) (x - 2), x = -1, 2
 (v) p(x) = x² , x = 0

(viii) p(x) = 2x +  1, x = 1/2

Answer

(i) If x = -1/3 is a zero of polynomial p(x) = 3x + 1 then p(-1/3) should be 0.
At, p(-1/3) = 3(-1/3)  1 = -1  1 = 0
Therefore, x = -1/3 is a zero of polynomial p(x) = 3x + 1.

(ii) If x = 4/5 is a zero of polynomial p(x) = 5x - π then p(4/5) should be 0.
At, p(4/5) = 5(4/5) - π = 4 - π
Therefore, x = 4/5 is not a zero of given polynomial p(x) = 5x - π.
 

(iii) If x = 1 and x = -1 are zeroes of polynomial p(x) = x² - 1, then p(1) and p(-1) should be 0.
At, p(1) = (1)² - 1 = 0 and
At, p(-1) = (-1)² - 1 = 0
Hence, x = 1 and -1 are zeroes of the polynomial  p(x) = x² - 1.

(iv) If x = -1 and x = 2 are zeroes of polynomial p(x) = (x + 1) (x - 2), then p( - 1) and (2) should be 0.
At, p(-1) = (-1 + 1) (-1 - 2) = 0 (-3) = 0, and
At, p(2) = (2 + 1) (2 - 2) = 3 (0) = 0
Therefore, x = -1 and x = 2 are zeroes of the polynomial p(x) = (x + 1) (x - 2).

(v) If x = 0 is a zero of polynomial p(x) = x², then p(0) should be zero.
Here, p(0) = (0)² = 0
Hence, x = 0 is a zero of the polynomial p(x) = x².

(viii)  If x = 1/2 is a zero of polynomial p(x) = 2x +  1 then p(1/2) should be 0.
At, p(1/2) = 2(1/2) 1 = 1 + 1 = 2
Therefore, x = 1/2 is not a zero of given polynomial p(x) = 2x +  1.

4. Find the zero of the polynomial in each of the following cases:
 (i) p(x) = x + 5 

(ii) p(x) = x - 5 

(iii) p(x) = 2x + 5
 (iv) p(x) = 3x - 2 

(v) p(x) = 3x 

(vi) p(x) = ax, a ≠ 0
 (vii) p(x) = cx  d, c ≠ 0, c, are real numbers.

Answer

Exercises 2.3

 1. Find the remainder when x³ + 3x²  + 3x + 1 is divided by
 (i) x + 1
 (ii) x - 1/2
 (iii) x
 (iv) x + π
 (v) 5 + 2x

Answer

(i) x + 1
By long division,

Therefore, the remainder is 0.

(ii) x - 1/2
By long division,

Therefore, the remainder is 27/8.

(iii) x

Therefore, the remainder is 1.

(iv) x + π

Therefore, the remainder is [1 - 3π 3π² - π³].

(v) 5 + 2x

Therefore, the remainder is -27/8.

2. Find the remainder when x³ - ax² + 6x - a is divided by x - a.

Answer

By Long Division,

Therefore, remainder obtained is 5a when x³ - ax² +  6x - a is divided by x - a.

3. Check whether 7 + 3x is a factor of 3x³ + 7x.

Answer

We have to divide 3x³ + 7x by 7 + 3x. If remainder comes out to be 0 then 7 3x will be a factor of 3x³ + 7x.
By Long Division,

As remainder is not zero so 7 + 3x is not a factor of 3x³ + 7x.