CBSE CLASS 10 POLYNOMIALS

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 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.

Answer2:

http://img0.meritnation.com/img/curr/1/10/9/129/1987/Chapter%202_html_5ae45263.gif

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

http://img0.meritnation.com/img/curr/1/10/9/129/1987/Chapter%202_html_m5c14590b.gif

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 http://img0.meritnation.com/img/curr/1/10/9/129/1917/Chapter%202_html_m6f1b144a.gif

Therefore, the zeroes of 6x2 − 3 − 7x arehttp://img0.meritnation.com/img/curr/1/10/9/129/1917/Chapter%202_html_ee4f088.gif.

Sum of zeroes = http://img0.meritnation.com/img/curr/1/10/9/129/1917/Chapter%202_html_15d94173.gif

Product of zeroes = http://img0.meritnation.com/img/curr/1/10/9/129/1917/Chapter%202_html_11f40d6e.gif

 

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 http://img0.meritnation.com/img/curr/1/10/9/129/1917/Chapter%202_html_m6f1b144a.gif

Therefore, the zeroes of 6x2 − 3 − 7x arehttp://img0.meritnation.com/img/curr/1/10/9/129/1917/Chapter%202_html_ee4f088.gif.

Sum of zeroes = http://img0.meritnation.com/img/curr/1/10/9/129/1917/Chapter%202_html_15d94173.gif

Product of zeroes = http://img0.meritnation.com/img/curr/1/10/9/129/1917/Chapter%202_html_11f40d6e.gif

 

 

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

Answer2:

http://img0.meritnation.com/img/curr/1/10/9/129/1987/Chapter%202_html_5ae45263.gif

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

http://img0.meritnation.com/img/curr/1/10/9/129/1987/Chapter%202_html_m5c14590b.gif

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