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Question-1

Find all the common zeroes of the polynomials: x3 + 5x2 – 9x – 45 and x3 + 8x2 + 15x.

Solution:
Let P(x) = x3 + 5x2 – 9x – 45 and Q (x) = x3 + 8x2 + 15x.
Let P(x) = x3 + 5x2 – 9x – 45
= x2(x + 5) – 9(x + 5)
                                            = (x + 5) (x2 – 9)
                                            = (x + 5) (x + 3) (x – 3)
Q (x) = x3 + 8x2 + 15x
= x (x2 + 8x + 15)
                               = x (x + 5) (x + 3)
The common zeroes of the polynomials are (x + 5) and (x + 3).

Question-2

Determine whether the given value of x is a zero of the polynomial 3x2 - 2x - 1 or not; if x = 1

Solution:
Let p(x) = 3x2 - 2x -1
   p(1) = 3(1)2 - 2(1) - 1 = 3 - 2 - 1 = 0. 
Therefore the given value of x is a zero of the given polynomial.

Question-3

Determine whether the given value of x is a zero of the given polynomial or not: 2x2 - 6x + 3; x = 

Solution:
Let p(x) = 2x2 - 6x + 3
  p() = 2()2 - 6() + 3 = - 3 + 3 =
Therefore the given value of x is not a zero of the given polynomial.

Question-4

Determine whether the given value of x is a zero of the given polynomial or not: (2x + 3)(3x - 2); x = 

Solution:
Let p(x) = (2x + 3) (3x - 2)
p()=p( (2 × + 3)(3 × - 2) = ( + 3)(2 - 2) = ( + 3) 0 = 0. 
Therefore the given value of x is a zero of the given polynomial

Question-5

Determine whether the given value of x is a zero of the given polynomial or not: x2 + x + 1; x = -1

Solution:
Let p(x) = x2 + x + 1
(-1) = (-1)2 + (-1) + 1 = 1 – 1 + 1 = 1. 
Therefore the given value of x is not a zero of the given polynomial

Question-6

Determine whether the given values of x are zeroes of the given polynomial or not: x2 + 6x + 5; x = -1, x = -5

Solution:
Let p(x) = x2 + 6x + 5;
Put x = -1
p(-1)= (-1)2 + 6(-1) + 5 = 1 – 6 + 5 = 0
 
Put x = -5
p(-5)=(-5)2 + 6(-5) + 5 = 25 - 30 + 5 = 0

Therefore the given value of x is a zeroes of the given polynomial.

Question-7

In the following, determine whether the given values of x are zeroes of the given polynomial or not: 6x2 - x - 2; x = -, x = 

Solution:
Let p(x) = 6x2 - x -2
Put x = -
p(
-) = 6(-)2 - (-) - 2 = + - 2 = = 0

Put x = 
p() = 6x2 - x -2 = 6()2 - () - 2 = - - 2 = =

Therefore the given value of x = - is a zero and x = is not the zero of the given polynomial.

Question-8

In the following, determine whether the given values of x are zeroes of the given polynomial or not:
x2 +; x =, x = -2

Solution:
Let p(x) = x2
Put x = 
p() = x2 + = ()2 + = 2 + 2 - 4 = 0
Put x = -2
p( -2) = x2 + = (-2)2 + = 8 – 4 – 4 = 0
Therefore the given value of x is a zero of the given polynomial.

Question-9

Determine whether the given values of x are zeroes of the given polynomial or not:
9x2-3x-2; x = -, x =

Solution:
Let p(x) = 9x2-3x-2
Put x = - 
⇒p(-) = 9(-)2 - 3(-) – 2 = 1 + 1 – 2 = 0
Put x =
⇒p() = 9x2 - 3x - 2 = 9()2 - 3() – 2 = 4 - 2 – 2 = 0
Therefore the given value of x is a zero of the given polynomial.

Question-10

Determine whether the given values of x are zeroes of the given polynomial or not:(x+4)(x-5); x = -4, x=5

Solution:
Let p(x) = (x+4)(x-5)
Put x = -4
⇒p(-4) = (x + 4)(x - 5) = (-4 + 4)(-4 - 5) = 0(-9 ) = 0

Put x = 5


⇒p(5) =  (x + 4)(x - 5) = (5 + 4)(5 - 5) = 9(0) = 0

Therefore the given value of x is a zero of the given polynomial.

Question-11

Determine whether the given values of x are zeroes of the given polynomial or not: (3x+8)(2x+5); x = 2, x = 2

Solution:
Let p(x) = (3x+8)(2x+5)
Put x = 2 =
⇒p()  = (3 × + 8)(2 × + 5) = (8 + 8)( + 5) = 16 × =
Put x = 2 =
⇒p(  ) = (3 × + 8)(2 × + 5) = ( +8)(5+5) = × 10 = 155
Therefore the given value of x is not the zero of the given polynomial.

Question-12

Find the sum and product of the zeroes of the polynomial x2 - 6x + 5;

Solution:
x2 - 6x + 5
Sum of the zeroes of the polynomial = = = 6
Product of the zeroes of the polynomial = = = 5

Question-13

Find the sum and product of the zeroes of the polynomial px2 + qx + pq.

Solution:
px2 + qx + pq

Sum of the zeroes of the polynomial = =
Product of the zeroes of the polynomial = = = q.

Question-14

Find the sum and product of the zeroes of the polynomial x2 – 25.

Solution:
x2 – 25
Sum of the zeroes of the polynomial = = = 0 Product of the zeroes of the polynomial = = = - 25.

Question-15

Find the sum and product of the zeroes of the polynomial 4x2 - 7x.

Solution:
4x2 - 7x
Sum of the zeroes of the polynomial = = = Product of the zeroes of the polynomial = = = 0

Question-16

Form the polynomial whose zeroes are 5, 6.

Solution:
The roots are 5 and 6. Sum of the zeroes= 5 + 6 = 11
Product of the zeroes = 5 × 6 = 30

The required polynomial is x2 - (sum of the roots) x + Product of the zeroes

x2 - 11x + 30 is the required polynomial.


Question-17

Form the polynomial whose zeroes are 2, -2.

Solution:
The zeroes are 2 and -2.
Sum of the zeroes = 2 + (-2) = 0
Product of the zeroes = 2 × (-2) = - 4

The required polynomial is x2 - (sum of the zeroes)x + Product of the zeroes
 
x2 - (0) x + (- 4)
x2 – 4

Question-18

Form the polynomial whose zeroes are 3 + , 3 - .

Solution:
 The zeroes are 3 + and 3 - . Sum of the zeroes = (3 + ) + (3 - )= 6 Product of the zeroes = (3 + ) (3 - ) = 9 - ()2
                                                                    = 9 – 3 = 6

\ The required polynomial is
x2 - (sum of the zeroes) x + Product of the zeroes x2 - (6) x + (6)
x2 - 6x + 6

Question-19

Form the polynomial whose zeroes are .

Solution:
The zeroes are and . Sum of the zeroes = +
                             =
                             = = 4

Product of the zeroes = ×
                                 =
                                         =
                                 =
                                 =
                                 =

∴ The required polynomial is x2 - (sum of the zeroes) x + Product of the zeroes
x2 - (4) x + () Þ 2x2 - 8x + 7

Question-20

If α and β are the roots of the polynomial ax2 + bx + c, then find the value of α 2 + β 2

Solution:
The polynomial is ax2 + bx + c whose zeroes are α and β .

The sum of the zeroes =
α + β =
The product of the zeroes =
α β =

α 2 + β 2 = (α + β )2 - 2α β = ()2 – 2 ×
                                             =
                                             =

Question-21

Form the polynomial whose zeroes are 3 +, 3 - .

Solution:
The zeroes are 3 + , 3 - . Sum of the zeroes = (3 +  ) + (3 -  ) = 6 Product of the zeroes = (3 +  ) (3 -  )
                                  = 9 - (
)2
                                  = 9 – 3
                                  = 6


∴ The required polynomial is
x2 - (sum of the zeroes) x + Product of the zeroes x2 - (6)x + (6) Þ x2 - 6x + 6

Question-22

Form the polynomial whose zeroes are .

Solution:






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