Polynomials – Exercise 2.2 – Class IX

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

(i) x = 0

(ii) x = -1

(iii) x = 2

Solution:

P(x) = 5x – 4x2 + 3

(i) x = 0 , P(0) = 5(0) – 4(0)2 + 3 = 0 – 0 + 3 = 3

(ii) x = -1, P(-1) = 5(-1) – 4(-1)2 + 3 = -5 – 4 + 3 = -9 + 3 = -6

(iii) x = 2, P(2)= 5(2)– 4(2)2 + 3 = 10 – 16 + 3 = – 3


  1. 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)

Solution:

(i) p(y) = y2 – y + 1

p(0) = 02 – 0 + 1 = 1

p(1) = 12 – 1 + 1 = 1

p(2) = 22 – 2 + 1 = 4 – 2 + 1 = 3

 

(ii) p(t) =2 + t + 2t2 – t3

p(0) = 2 + (0) + 2(0)2 – (0)3 = 2

p(1) = 2 + (1) + 2(1)2 – (1)3 = 2 + 1 + 2 – 1 = 4

p(2) = 2 + (2) + 2(2)2 – (2)3 = 2 + 2 + 8 – 8 = 4

 

(iii) p(x) = x3

p(0) = 0

p(1) = 13 = 1

p(2) = 23 = 8

 

(iv) p(x) = (x – 1)(x + 1)

p(0) = (0 – 1)(0 + 1) = -1 x 1 = -1

p(1) = (1 – 1)(1 + 1) = 0x2 = 0

p(2) = (2 – 1)(2 + 1) = 1 x 3 = 3


  1. 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

(vi) p(x) = lx + m , x = –m/l

(vii) p(x) = 3x2 – 1 , x = – 1/√3, 2/√3

(viii) p(x) = 2x + 1, x = 1/2

Solution:

(i) p(x) = 3x + 1 , x = –1/3

Yes zeroes of the polynomial. 3x + 1 = 0 for x = –1/3

(ii) p(x) = 5x – π , x = 4/5

No, 5x – π = 5(4/5) – π= 4 – π ≠ 0 at x = 4/5

(iii) p(x) = x2 – 1 , x = 1, – 1

Yes. x2 – 1 = 12 – 1 = 0 at x = 1,

x2 – 1 = (-1)2 – 1 = 1 – 1 = 0 at x = – 1

(iv) p(x) = (x + 1)(x – 2), x = -1, 2

yes.

(x + 1)(x – 2) = (-1 + 1)(-1 – 1) at  x = -1,

(x + 1)(x – 2) = (2 + 1)(2 – 2) = 3×0 = 0 at  x = 2

 

(v) p(x) = x2 , x = 0

Yes. x2 = 02 = 0

 

(vi) p(x) = lx + m , x = –m/l

Yes. l(-m/l) + m = -m + m = 0 at x = –m/l

 

(vii) p(x) = 3x2 – 1 , x = – 1/√3, 2/√3

3x2 – 1 = 3(- 1/√3)2 – 1 = 3(1/3) – 1 = 0  at x = – 1/√3

3x2 – 1 = 3(- 2/√3)2 – 1 = 3(4/3) – 1 = 4 – 1 = 3≠0  at –2/√3

Therefore, at – 1/√3 is a zero but –2/√3 is not a zero of the polynomial.

(viii) p(x) = 2x + 1, x = 1/2

2x + 1 = 2(1/2) + 1 = 2≠0 at x = 1/2


  1. 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 , d are real numbers

Solution:

(i) p(x) = x + 5

x + 5 = 0 so, x = -5

Therefore, -5 is the zero of x + 5

(ii) p(x) = x – 5

x – 5 = 0 so, x = 5

Therefore, 5 is the zero of x – 5.

 

(iii) p(x) = 2x + 5

2x + 5 = 0 so, x = –5/2

Therefore, –5/2 is the zero of 2x + 5

 

(iv) p(x) = 3x – 2

3x – 2 = 0 so, x = 2/3

Therefore, 2/3 is the zero of 3x – 2

 

(v) p(x) = 3x

3x = 0 so, x = 0

Therefore, 0 is the zero of 3x

 

(vi) p(x) = ax , a ≠ 0

ax = 0 so, a = 0/x

Therefore, 0/x is the zero of ax

 

(vii) p(x) = cx + d , c≠0, c , d are real numbers

cx + d = 0 so, x = –d/c

Therefore, –d/c is the zero of cx + d


 

 

 

 

 

 

 

 

 

 

 

Polynomial – Exercise 2.1 – Class IX

  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

Solution:

(i)Polynomial in one variable, x

(ii)Polynomial in one variable, y

(iii) 3√t + t√2 is not  a  polynomial as power of t in √2t is  not a whole number

(iv) y + 2/y is not a polynomial as a power of  y  in 1/y = y-1  is not a whole number

(v) x10 +y3 +  t50 is not a polynomial with three variables x , y and t.


  1. Write the coefficients of x2 in each of the following:

(i) 2+ x2 + x

(ii) 2 – x2 + x3

(iii) π/2 . x2 + x

(iv)√2x – 1

Solution:

(i) 2+ x2 + x, coefficient of x2 is 1

(ii) 2 – x2 + x3, coefficient of x2 is -1

(iii) π/2 . x2 + x, coefficient of x2 is π/2

(iv)√2x – 1, x2 is not there therefore there is no coefficient.


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

Solution:

Binomial of degree 35  can be a35 + 10

monomial of degree 100 can be x100


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

(i) 5x3 + 4x2+ 7

(ii)  4 – y2

(iii)   5t – √7

(iv)  3

Solution:

(i) 5x3 + 4x2+ 7, degree of a polynomial is 3 as x3 is the highest power.

(ii)  4 – y2, degree of a polynomial is 2 as x2 is the highest power.

(iii)   5t – √7, degree of a polynomial is 1 as x is the highest power.

(iv)  3, degree of a polynomial is 0 as x0 is the highest power.


  1. Classify the following as linear, quadratic and cubic polynomials:

(i)x2 + x

(ii) x – x3

(iii)y + y2 + 4

(iv) 1 + x

(v) 3t

(vi) r2

(vii) 7x3

Solution:

(i)x2 + x, is quadratic

(ii) x – x3, , is cubic

(iii)y + y2 + 4, is quadratic

(iv) 1 + x, is linear

(v) 3t, is linear

(vi) r2, is quadratic

(vii) 7x3, is cubic


 

 

Polynomial

A polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

An example of a polynomial of a single indeterminate a is a2 − 4a + 10.

Also see



 

Cubic Polynomial

A cubic polynomial is a polynomial of degree 3.  An equation involving a cubic polynomial is called a cubic equation.

Example: a³ − 4a² + 10a + 5

 

Linear Polynomial

A linear polynomial is the same thing as a degree 1 polynomial.

Ex:

(i) 1 + x

(ii) x + 10000 etc

 

How to find rational numbers between any two numbers

To find a rational number between any two numbers such as r and s, you can add r and s and divide the sum by 2,  ie..,  (r+s)/2  lies between r and s.

Example:

Let r = 1 and s = 2. Find 6 rational numbers between 1 and 2.

We literally want rational numbers i.e., fractions between 1 and 2. The resultant fraction must be greater than 1 and less than 2.

How to find rational numbers between any two numbers

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Quadratic Polynomial

A quadratic polynomial is a polynomial of degree 2.

An equation involving a quadratic polynomial is called a quadratic equation.

Example:

  1.  x2 + x
  2. y + y2 + 4
  3. r