UP Board Solutions for Class 10 Maths Chapter 2 Exercise 2.2 Polynomials

EXERCISE 2.2

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients:

(i) x2 – 2x – 8                (ii) 4s2 – 4s + 1

(iii) 6x2 – 3 – 7x            (iv)4u2 +8u                                    

(v) t2 – 15                    (vi) 3x2 – x – 4   

                            

Sol. (i) We have,   x2 – 2x – 8  = x2 – 2x – 4x-8

                                               = x(x + 2) – 4(x + 2)

                                                 = (x + 2)(x – 4)

The value of  x2 – 2x – 8  is zero when the value of (x + 2)(x – 4) is zero, i.e., when        

        x + 2 = 0 or x – 4 = 0, i.e.,

 when x = – 2 or x = 4.

          • The zeroes of x2 – 2x – 8    are – 2 and 4.

             Therefore, sum of the zeroes = (-2) + 4 = 2

and,               product of zeroes = (- 2)(4) = – 8 =

                                                   

(ii) We have, 4s2 – 4s + 1 =-4s2 – 2s – 2s+1

                                                = 2s(2s – 1) – 1(2s – 1)

                                                = (2s – 1)(2s – 1)

The value of 4s2 – 4s + 1 is zero when the value of (2s – 1)(2s – 1) is zero, i.e.,

when 2s – 1 = 0 or 2s – 1 = 0,

i.e., when s = or s =

=The zeroes of 4s2 – 4s + 1 are   and

Therefore, sum of the zeroes =  + = 1

and,      product of zeroes 

      

 (iii) We have,6x2 – 3 – 7x =6x2 – 7x – 3

                                              = 6x2– 9x + 2x – 3

                                            =3х(2x – 3) + 1(2x – 3)

                                              =(3x + 1)(2x – 3)

          The value of 6x2 – 3 – 7x  is zero when the value of

          (3x + 1)(2x – 3) is zero, i.e., when 3x + 1 = 0 or 2x – 3 = 0,

          i.e., when x = –  or x =

         => The zeroes of  6x2 – 3 – 7x  are –   and

        Therefore, sum of the zeroes

         and,        product of  zeroes 

 (iv) We have, 4u2 +8u 4 = 4u(u + 2)

         The value of 4u2 +8u   is zero when the value of 4u(u + 2) is zero, i.e., when u = 0

          or   u + 2 = 0, i.e., when  u = 0 or u = – 2.

         => The zeroes of 4u2 +8u are 0 and – 2.

               sum of the zeroes = 0 ÷ (-2) =- 2

                                              

             

                                             

 (v) We have, t2 – 15

   

 (vi) We have, 3x2 – x – 4   =3x2 -3 x – 4x- 4

                                             = 3x(x + 1) – 4(x + 1)

                                             = (x + 1)(3x – 4)

         The value of 3x2 – x – 4  is zero when the value of (x + 1)(3x – 4) is zero, i.e.,

         when x + 1 = 0 or 3x – 1 = 0, i.e.,

2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

         

Sol. (i) Let the polynomial be ax2 + bx + c, and its zeroes be a and . Then,

    => One quadratic polynomial which fits the given conditions is x2 – o.x +  , i.e., x 2 + .

(iv) Let the polynomial be ax? + bx + c, and its zeroes be a and .

Then, a + = 1

a = 1

If a = 1, then b = -1and c = 1.

=> One quadratic polynomial which fits the given conditions is -x +1

(v)Let the polynomial be ax2 + bx + c and its zeroes be a and .

Then,

If a = 4, then b = -1and c = 1.

=> One quadratic polynomial which fits the given conditions is 4x2 -x + 1.

(vi) Let the polynomial be ax? + bx + c and its zeroes be a and B. Then,

If a = 4, then b = -4and c = 1.

=> One quadratic polynomial which fits the given conditions is x2 -4x + 1.

                                                  

                                                 

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