# Question-1

**Find all the common zeroes of the polynomials: x**

^{3}+ 5x^{2}â€“ 9x â€“ 45 and x^{3}+ 8x^{2}+ 15x.**Solution:**

Let P(x) = x

^{3}+ 5x

^{2}â€“ 9x â€“ 45 and Q (x) = x

^{3}+ 8x

^{2}+ 15x.

Let P(x) = x

^{3}+ 5x

^{2}â€“ 9x â€“ 45 = x

^{2}(x + 5) â€“ 9(x + 5)

= (x + 5) (x

^{2}â€“ 9)

= (x + 5) (x + 3) (x â€“ 3)

Q (x) = x

^{3}+ 8x

^{2}+ 15x = x (x

^{2}+ 8x + 15)

= x (x + 5) (x + 3)

âˆ´ 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 3x**

^{2 }- 2x - 1 or not; if x = 1**Solution:**

Let p(x) = 3x

^{2 }- 2x -1

â‡’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: 2x**

^{2 }- 6x + 3; x =**Solution:**

Let p(x) = 2x

^{2 }- 6x + 3

â‡’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)

â‡’p()

**=**p(

**(2 Ã— + 3)(3 Ã— - 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: x**

^{2 }+ x + 1; x = -1**Solution:**

Let p(x) = x

^{2 }+ x + 1

â‡’(-1) = (-1)

^{2 }+ (-1) + 1 = 1 â€“ 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: x**

^{2 }+ 6x + 5; x = -1, x = -5**Solution:**

Let p(x) = x

^{2 }+ 6x + 5;

Put x = -1

â‡’p(-1)= (-1)

^{2 }+ 6(-1) + 5 = 1

^{ }â€“ 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: 6x**

^{2 }- x - 2; x = -, x =**Solution:**

Let p(x) = 6x

^{2 }- x -2

Put x = -

p(-) = 6(-)

^{2 }- (-) - 2 = + - 2 = = 0

Put x =

p() = 6x

^{2 }- 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:**

x

x

^{2 }+; x =, x = -2**Solution:**

Let p(x) = x

^{2 }+

Put x =

â‡’p() = x

^{2 }+ = ()

^{2 }+ = 2 + 2 - 4 = 0

Put x = -2

p( -2) = x

^{2 }+ = (-2)

^{2 }+ = 8 â€“ 4 â€“ 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:**

9x

9x

^{2}-3x-2; x = -, x =**Solution:**

Let p(x) = 9x

^{2}-3x-2

Put x = -

â‡’p(-) = 9(-)

^{2 }- 3(-) â€“ 2 = 1 + 1 â€“ 2 = 0

Put x =

â‡’p() = 9x

^{2 }- 3x - 2 = 9()

^{2 }- 3() â€“ 2 = 4 - 2 â€“ 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

â‡’p(-4) = (x + 4)(x - 5) = (-4 + 4)(-4 - 5) = 0(-9 ) = 0

Put x = 5

â‡’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 =

â‡’p() = (3 Ã— + 8)(2 Ã— + 5) = (8 + 8)( + 5) = 16 Ã— =

Put x = 2 =

â‡’p( ) = (3 Ã— + 8)(2 Ã— + 5) = ( +8)(5+5) = Ã— 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 x**

^{2 }- 6x + 5;**Solution:**

x

^{2 }- 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 px**

^{2 }+ qx + pq.**Solution:**

px

^{2 }+ 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 x**

^{2 }â€“ 25.**Solution:**

x

^{2 }â€“ 25

âˆ´ Sum of the zeroes of the polynomial = = = 0 âˆ´ Product of the zeroes of the polynomial =

**=**

**= - 25.**

# Question-15

**Find the sum and product of the zeroes of the polynomial 4x**

^{2 }- 7x.**Solution:**

4x

^{2 }- 7x

âˆ´ Sum of the zeroes of the polynomial = = = âˆ´ 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. âˆ´ Sum of the zeroes= 5 + 6 = 11

âˆ´ Product of the zeroes = 5 Ã— 6 = 30

The required polynomial is x

^{2}- (sum of the roots) x + Product of the zeroes

â‡’ x^{2 }- 11x + 30 is the required polynomial.

# Question-17

**Form the polynomial whose zeroes are 2, -2.****Solution:**

The zeroes are 2 and -2.

âˆ´ Sum of the zeroes = 2 + (-2) = 0

âˆ´ Product of the zeroes = 2 Ã— (-2) = - 4

The required polynomial is x

^{2}- (sum of the zeroes)x + Product of the zeroes

â‡’ x

^{2 }- (0) x + (- 4)

\ x

^{2 }â€“ 4

# Question-18

**Form the polynomial whose zeroes are 3 + , 3 - .****Solution:**

The zeroes are 3 + and 3 - . âˆ´ Sum of the zeroes = (3 + ) + (3 - )= 6 âˆ´ Product of the zeroes = (3 + ) (3 - ) = 9

**-**()

^{2}

= 9 â€“ 3 = 6

\ The required polynomial is

x

^{2}- (sum of the zeroes) x + Product of the zeroes â‡’ x

^{2 }- (6) x + (6)

âˆ´ x

^{2 }- 6x + 6

# Question-19

**Form the polynomial whose zeroes are , .****Solution:**

The zeroes are and

**.**âˆ´ Sum of the zeroes = +

=

**=**= 4

âˆ´ Product of the zeroes =

**Ã—**

=

**=**

**=**

=

=

âˆ´ The required polynomial is x

^{2}- (sum of the zeroes) x + Product of the zeroes

â‡’ x

^{2 }- (4) x + () Ãž 2x

^{2 }- 8x + 7

# Question-20

**If Î± and Î² are the roots of the polynomial ax**^{2 }+ bx + c, then find the value of Î±^{2}+ Î²^{2 }**Solution:**

The polynomial is ax

^{2 }+ bx + c whose zeroes are Î± and Î² .

The sum of the zeroes = Î± + Î² =

The product of the zeroes = Î± Î² =

âˆ´ Î±

^{2}+ Î²

^{2 }= (Î± + Î² )

^{2 }- 2Î± Î² = ()

^{2 }â€“ 2 Ã—

**=**â€“

**=**

# Question-21

**Form the polynomial whose zeroes are 3 +, 3 -****.****Solution:**

The zeroes are 3 + , 3 - . âˆ´ Sum of the zeroes = (3 + ) + (3 - ) = 6 âˆ´ Product of the zeroes = (3 + ) (3 - )

= 9

**-**()

^{2}

= 9 â€“ 3

= 6

âˆ´ The required polynomial is

x

^{2}- (sum of the zeroes) x + Product of the zeroes â‡’ x

^{2 }- (6)x + (6) Ãž x

^{2 }- 6x + 6

# Question-22

**Form the polynomial whose zeroes are**

**.****Solution:**