# 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.

= 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 2x23x+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.

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.

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

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.

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

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

= 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

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