# CBSE CLASS 10 POLYNOMIALS

# Polynomials#1 Ex2.1 Explained – CBSE Class 10 Math

# Polynomials#2 Ex2.2 Solved & Explained – CBSE Class 10 Math in Hindi

# Polynomials#3 Ex2.3 (Q.No-1,2) – CBSE Class 10 Math

# Polynomials#4 Ex2.3 (Q.No-3,4) – CBSE Class 10 Math

POLYNOMIALS

BASIC CONCEPT AND FORMULAS

ONE –MARK Questions

1.If zeros of the polynomial x2-4x+1 is 2+ , write the other zero.

SOL: Let the other zero = α .

Now, (2+ )+α= , => α=4-2-=2-.

2.Find the zeroes of the polynomial 2

SOL Let p(x)= 2

Now p(x)= 0=>2 =0

=>2

=> ~~ ~~

=> x =

3. The following figure shows the graphs of y = p(x), where p(x) is a polynomial.

Find the number of zeroes of the polynomial.

SOL .Number of zeroes = 3 .

4.Find a quadratic polynomial, the sum and product of whose zeroes are –3 and 2, respectively.

SOL Quadratic polynomial=k(

= k( )

= ( ),k=1

5.Find the quadratic polynomial whose zeroes are~~ and -4~~

SOL : Quadratic polynomial=k( )

= k(

= (,k=1

**6. p(x)=ax2+bx+c.If a+b+c=0, then find one of its zeroes.**

Ans: It is given that a+b+c=0,

p(x)=ax2+bx+c

p(1)=a(1)2+b(1)+c

=a+b+c

=0.

So x=1 is a zero of the polynomial.

TWO MARKS QUESTIONS

1.Find the quadratic polynomial whose zeroes are and

SOL : Quadratic polynomial=k( )

=k(

.

2. If one zero of 2 – 3x + k is reciprocal to the other, then find the value of k .

Sol; Let one zero=p other zero=

So,p× = => k=2

3.. If ~~ are the zeroes of the quadratic polynomial , find the value of a if ~~

Sol; ~~=6 =>2~~~~=12~~

On solving ~~, ~~

Now,~~=~~~~=8×2=16~~

4.If are the zeroes of the polynomial find the value of k.

Sol; =

=> k=8

5. The graph of y = f(x), where f(x) is a polynomial in x is given below .Find the number of zeroes lying

between –2 to 0.

Sol; number of zeroes=3

6.If (x + a) is a factor of 2 + 2ax + 5x + 10, find a.

Sol; Let p(x)= 2 + 2ax + 5x + 10

p(-a)= 2~~ 2+ 2a~~~~ + 5 ~~~~ + 10~~

=> 0 = 2 – 2 -5a+10

=> 5a=10

=> a=2

THREE MARKS QUESTIONS

1.If α and β are the zeroes of the quadratic polynomial f(x) = – 5x ~~6 then find the value of ~~

Sol; =

=

2. Give examples of polynomial p(x), g(x), q(x) and r(x), which satisfy the division algorithm .

Sol; degp(x) = deg q(x)

p(x)=6 x-12

g(x)=6

q(x)= x-2

Here, degp(x) = deg q(x)=1

3.If the product of the zeroes of the polynomial ax2 -6x -6 is 4,then find the value of a. Also find the sum

of zeroes of the polynomial.

Sol;product of the zeroes=4

6 a = 4

a =

sum of zeroes = = = 9

4.If ~~ and ~~~~are the zeroes of the quadratic polynomial 2x2+5x+k, find the value of k such that~~

(~~+~~~~)2 ~~~~ =24~~

Sol; (~~+~~~~)2 ~~~~ = (2 – =24~~

=> – = 24

=> = 24

=>k =

6. If one root of the quadratic polynomial 2x2~~3x+p is 3,find the other root. Also, find the value of p.~~

Sol;

Let the other root=k

3+k =

So ,k = –

FOUR MARKS QUESTIONS

Question 1:

Obtain all other zeroes of 3x4 +6x3-2x2-10x-5 , if two of its zeroes are and –.

Answer 1: Let four zeroes are a,b,c ,d where a = & b = –

Sum of zeroes=a+ b+ c + d = = –2 => +-+c+d= -2 => c +d=-2

Product of zeroes = a .b .c. d = => –.c.d = => c .d= 1

=> d =

so, c + = -2

=> = –2 => c2 +2c+1 =0 =>(c+1)2 =0 => c= -1

=> d= -1

Question 2:

If the zeroes of polynomial x3-3x2+x+1 are a-b, a, a+b, find a and b.

Answer2:

Zeroes are a-b, a, a+b.

Comparing the given polynomial with px3+qx2+rx+t , we obtain

p = 1, q = −3, r = 1, t = 1

The zeroes are 1-b, 1, 1+b.

Question3 Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x , and verify the relationship between the zeroes and the coefficients.

Answer3:

The value of 6x2 − 3 − 7x is zero when 3x + 1 = 0 or 2x − 3 = 0, i.e., x= or

Therefore, the zeroes of 6x2 − 3 − 7x are.

Sum of zeroes =

Product of zeroes =

4. Find all the zeroes of 2x4 – 3x3 – 3x2 + 6x – 2 if that two of its zeroes are

Answer 4: Let four zeroes are a,b,c ,d where a = & b= –

Sum of zeroes = a+ b+ c + d = => +-+c+d = => c +d=

Product of zeroes = a .b .c. d = = -1=> . –.c.d = -1 => c .d=

=> d =

so, c + =

=> = =>2 c2 +3c+1 =0 => (c + 1) (2c + 1) = 0

5..Find all the zeroes of x3 -7x+6,if one of its zero is 1

Answer 5:

x-1 is factor of x3 -7x+6

By dividing we get x3 -7x+6 = (x-1)(x2+x-6)

Factorizing we get (x2+x-6 ) = (x-2)(x+3)

So two other roots are 2,-3.

6..Find a quadratic polynomial whose zeroes are 1 and -3. Now verify the relationship between zeroes and coefficients.

Answer6: Polynomial formed x2+2x-3

Sum of zeroes = 1+(-3) = -2

Product of zeroes = 1.(-3) = -3

Moreover,

So, sum of zeroes =

Product of zeroes =

MOST EXPECTED QUESTIONS

1. Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x , and verify the relationship between the zeroes and the coefficients.

Answer1:

The value of 6x2 − 3 − 7x is zero when 3x + 1 = 0 or 2x − 3 = 0, i.e., x= or

Therefore, the zeroes of 6x2 − 3 − 7x are.

Sum of zeroes =

Product of zeroes =

2.If the zeroes of polynomial x3-3x2+x+1 are a-b, a, a+b, find a and b

Answer2:

Zeroes are a-b, a, a+b.

Comparing the given polynomial with px3+qx2+rx+t , we obtain

p = 1, q = −3, r = 1, t = 1

The zeroes are 1-b, 1, 1+b.

3.If the product of the zeroes of the polynomial ax2 -6x -6 is 4,then find the value of a. Also find the sum of zeroes of the polynomial.

Sol; Product of the zeroes=4

6 a=4

a=

sum of zeroes===9

4.If α and β are the zeroes of the quadratic polynomial f(x) = – 5x ~~6 then find the value of ~~

Sol; =

=

5.Find the quadratic polynomial whose zeroes are and

: Quadratic polynomial=k( )

=k(

6.Find the quadratic polynomial whose zeroes are~~ and -4~~

SOL : Quadratic polynomial=k( )

= k(

7.Find a quadratic polynomial, the sum and product of whose zeroes are –3 and 2, respectively

SOL: Quadratic polynomial=k(

= k( )

8.Find the zeroes of the polynomial 2

SOL: Let p(x)= 2

Now p(x)= 0=>2 =0

=>2

=> ~~ ~~

=> x =

9.Find all the zeroes of x3 -7x+6,if one of its zero is 1

Answer:

x-1 is factor of x3 -7x+6

By dividing we get x3 -7x+6=(x-1)(x2+x-6)

Factorizing we get (x2+x-6)=(x-2)(x+3)

So two other roots are 2,-3.

10..Find a quadratic polynomial whose zeroes are 1 and -3. Now verify the relationship between zeroes and coefficients.

Answer: Polynomial formed x2+2x-3

Sum of zeroes= 1+(-3)=-2

Product of zeroes= 1.(-3)= -3

Moreover,

So, sum of zeroes =

Product of zeroes =

11. Find all the zeroes of 2x4 – 3x3 – 3x2 + 6x – 2 if that two of its zeroes are

Answer : Let four zeroes are a,b,c ,d where a = & b= –

Sum of zeroes=a+ b+ c + d = => +-+c+d= => c +d=

Product of zeroes = a .b .c. d= = -1=> . –.c.d = -1 => c .d=

=> d =

so, c + =

=> = =>2 c2 +3c+1 =0 => (c + 1) (2c + 1) = 0

=> c= -1 or

=> d= or -1