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**Ques 21: Find the zeroes of the quadratic polynomial 5x ^{2} - 4 - 8x and verify the relationship between the zeroes and the coefficients of the polynomial.Sol:** p (x) = 5x

= 5x

= 5x

= 5x (x - 2) + 2 (x - 2)

= (x - 2) (5x + 2)

âˆ´ zeroes of p (x) are 2 and

Relationship Verification

Sum of the zeroes

â‡’

â‡’

â‡’ 8/5 = 8/5

i.e., L.H.S. = R.H.S. â‡’ relationship is verified.

Product of the zeroes =

â‡’

â‡’

i.e., L.H.S. = R.H.S.

â‡’ The relationship is verified.**Ques 22: Find the quadratic polynomial, the sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.Sol:** âˆµ The quadratic polynomial p (x) is given by

x

âˆ´ The required polynomial is

= x

= x

To find zeroes:

âˆµ x

= x (x - 6) - 2 (x - 6)

= (x - 6) (x - 2)

âˆ´ The zeroes of p (x) are 6 and 2.

Sol:

Comparing it with Ax

A = (a

Let one of the zeroes = a

âˆ´ The other zero = 1/Î±

Now, Product of the zeroes

â‡’

â‡’ 6a = a^{2} âˆ’ 9 â‡’ a^{2} âˆ’ 6a + 9 = 0

â‡’ (a âˆ’ 3)^{2} =0 â‡’ a âˆ’ 3=0

â‡’ a = 3

Thus, the required value of a is 3.**Ques 24: If the product of zeroes of the polynomial ax ^{2} - 6x - 6 is 4, find the value of â€˜aâ€™**

âˆµ Product of zeroes =

but product of zeroes is given as 4

âˆ´ â‡’ âˆ’ 6 = 4 Ã— a

â‡’ â‡’

Thus, the required value of a is -3/2.**Ques 25: Find all the zeroes of the polynomial x ^{4} + x^{3} - 34x^{2} - 4x + 120, if two of its zeroes are 2 and - 2.**Here p (x) = x

Sol:

âˆµ The two zeroes of p (x) are 2 and - 2

âˆ´ (x - 2) and (x + 2) are factors of p (x)

â‡’ (x - 2) (x + 2) is a factor of p (x)

â‡’ x

Now, dividing p(x) by x

âˆµ Remainder = 0

âˆ´ p (x) = (x^{2} - 4) (x^{2} + x - 30)

i.e., x^{2} + x - 30 is also a factor of p (x).

âˆµ x^{2} + x - 30 = x^{2} + 6x - 5x - 30 = x (x + 6) - 5 (x + 6)

= (x + 6) (x - 5) = [x - (- 6)] [x - 5]

- 6 and 5 are also zeroes of p (x).

â‡’ All the zeroes of the given polynomial are : 2, - 2, 5 and - 6**Ques 26: Find all the zeroes of the polynomial 2x ^{4} + 7x^{3} - 19x^{2 }- 14x + 30, if two of its zeroes are âˆš2 and -âˆš2. **

âˆµ âˆš2

âˆ´ are the factors of p (x).

â‡’ i.e., x

Now, dividing p (x) by x

âˆ´ p (x) = (2x^{2} + 7x - 15) (x^{2} - 2)

[âˆµ Remainder = 0]

â‡’ 2x^{2} + 7x - 15 is a factor of p (x)

âˆµ 2x^{2} + 7x - 15 = 2x^{2} + 10x - 3x - 15

= 2x (x + 5) - 3 (x + 5)

= (2x - 3) (x + 5)

=

âˆ´ 3/2 and - 5 are zeroes of p (x)

are the zeroes of p (x).**Ques 27: Find the quadratic polynomial whose zeroes are 1 and - 3. Verify the relation between the coefficients and the zeroes of the polynomial.Sol: **âˆµ The given zeroes are 1 and - 3.

âˆ´ Sum of the zeroes = 1 + (- 3) = - 2

Product of the zeroes = 1 Ã— (- 3) = - 3

A quadratic polynomial p (x) is given by

x

âˆ´ The required polynomial is

x

â‡’ x

Verification of relationship

âˆµ Sum of the zeroes

âˆ´

â‡’âˆ’ 2= âˆ’ 2

i.e., L.H.S = R.H.S â‡’ The sum of zeroes is verified

âˆµ Product of the zeroes =

âˆ´

â‡’âˆ’ 3= âˆ’ 3

i.e., L.H.S = R.H.S â‡’ The product of zeroes is verified.**Ques 28: Find the zeroes of the quadratic polynomial 4x ^{2} - 4x - 3 and verify the relation between the zeroes and its coefficients.**

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

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

=

âˆ´ are zeroes of p (x).

Verification of relationship

âˆµ Sum of the zeroes =

âˆ´

â‡’ 2/2 = 1 â‡’ 1= 1

â‡’

2/2 = 1 â‡’ 1 = 1

i.e., L.H.S = R.H.S â‡’ Sum of zeroes is verified

Now, Product of zeroes =

â‡’

i.e., L.H.S = R.H.S â‡’ Product of zeroes is verified.**Ques 29: Obtain all other zeroes of the polynomial 2x ^{3} - 4x - x^{2} + 2, if two of its zeroes are âˆš2 and -âˆš2.**

âˆµ

âˆ´ and are the factors of p (x)

â‡’ is a factor of p (x)

â‡’ x

Now, Dividing p (x) by (x

â‡’ p (x) = (x^{2} - 2) (2x - 1)

âˆ´ 2x - 1 is also a factor of p (x)

i.e.,is another factor of px.

â‡’ 1/2 is another zero of p (x)**Ques 30: Find all the zeroes of x ^{4} - 3x^{3} + 6x - 4, if two of its zeroes are âˆš2 and - âˆš2.**p (x) = x

Sol:

âˆµ

âˆ´ x-

â‡’ is a factor of p (x).

â‡’ x

On Dividing p (x) by x

Since, remainder = 0

âˆ´ (x^{2} - 2) (x^{2} - 3x + 2) = p (x)

Now, x^{2} - 3x + 2 = x^{2} - 2x - x + 2

= x (x - 2) - 1 (x - 2) = (x - 1) (x - 2)

i.e., (x - 1) (x - 2) is a factor of p (x)

âˆ´ 1 and 2 are zeroes of p (x).

âˆ´ All the zeroes of p (x) are ,**âˆš**2 , **- âˆš**2,** **1 and 2.**Ques 31: Find a quadratic polynomial whose zeroes are - 4 and 3 and verify the relationship between the zeroes and the coefficients.Sol: **We know that:

P (x) = x

âˆµ The given zeroes are - 4 and 3

âˆ´ Sum of the zeroes = (- 4) + 3 = - 1

Product of the zeroes = (- 4) Ã— 3 = - 12

From (1), we have

x

= x

Comparing (2) with ax

a = 1, b = 1, c = - 12

âˆ´ Sum of the zeroes = -b/a

â‡’ (+ 3) + (- 4) = -1/1

i.e., L.H.S = R.H.S â‡’ Sum of zeroes is verified.

Product of zeroes = c/a

â‡’ 3 Ã— (- 4) = -12/1

â‡’ - 12 = - 12

i.e., L.H.S = R.H.S â‡’ Product of roots is verified.

Sol:

g (x) = 2x + 1

Now, dividing f (x) by g (x), we have:

Thus, The quotient = 3x2 + 5x - 2

remainder = 0**Ques 33: If the polynomial 6x ^{4} + 8x^{3} + 17x^{2} + 21x + 7 is divided by another polynomial 3x^{2} + 4x + 1 then the remainder comes out to be ax + b, find â€˜aâ€™ and â€˜bâ€™.**We have:

Sol:

âˆ´ Remainder = x + 2

Comparing x + 2 with ax + b, we have

a = 1 and b = 2

Thus, the required value of a = 1 and b = 2.**Ques 34: If the polynomial x ^{4} + 2x^{3} + 8x^{2} + 12x + 18 is divided by another polynomial x^{2} + 5, the remainder comes out to be px + q. Find the values of p and q.**We have:

Sol:

âˆ´ Remainder = 2x + 3

Comparing 2x + 3 with px + q, we have

p = 2 and q = 3**Ques 35: Find all the zeroes of the polynomial x ^{3} + 3x^{2} - 2x - 6, if two of its zeroes are - âˆš2 and âˆš2.**p (x) = x

Sol:

âˆµ Two of its zeroes are -âˆš2 and âˆš2

â‡’ is a factor of p (x)

â‡’ x

Now, dividing p (x) by x

âˆ´ p (x) = (x^{2} - 2) (x + 3)

i.e., (x + 3) is a factor of p (x),

â‡’ (- 3) is a zero of p (x)

âˆ´All the zeroes of p (x) are **-** âˆš2, âˆš2 and - 3.**Ques 36: Find all the zeroes of the polynomial 2x ^{3} + x^{2} - 6x - 3, if two of its zeroes are -âˆš3 and âˆš3.** p (x) = 2x

Sol:

Two of its zeroes are -âˆš3

âˆ´ and are factors of p (x)

i.e., is a factor of p (x)

â‡’ x

Now, Dividing p (x) by x

âˆ´ p (x) = (x^{2} - 3) (2x + 1)

â‡’ is a factor of p (x)

â‡’ -1/2 is a zero of p (x)

âˆ´ All the zeroes of p (x) are -âˆš3 , âˆš3 and -1/2.**Ques 37: Find the zeroes of the polynomial and verify the relation between the coefficients and the zeroes of the above polynomial.****Sol:** The given polynomial is

âˆ´ zeroes of the given polynomial are

Now in,

co-efficient of x^{2} = 1

co-efficient of x = 1/6

constant term = â€“2

âˆ´ Sum of zeroes

Product of zeroes **Ques 38: Find the quadratic polynomial, the sum and product of whose zeroes are respectively. Also find its zeroes. ****Sol: **Sum of zeroes = âˆš2

Product of zeroes

âˆµ A quadratic polynomial is given by

x^{2} â€“ [sum of roots] x + [Product of roots]

âˆ´ The required polynomial is

â‡’

Since =

â‡’ zeroes are **Ques 39: If the remainder on division of x ^{3} + 2x^{2} + kx + 3 by x - 3 is 21, then find the quotient and the value of k. Hence, find the zeroes of the cubic polynomial x^{3} + 2x^{2} + kx - 18.**

âˆµ The divisor = x â€“ 3

âˆ´ p(3) = 3

21 = 27 + 18 + 3k + 3

[âˆµ Remainder = 21]

â‡’ 21 â€“ 18 â€“ 3 â€“ 27 = 3k

â‡’ â€“27 = 3k â‡’ k = â€“ 9

Now, the given cubic polynomial

= x

since,

âˆ´ The required quotient = x^{2} + 5x + 6

Now, x^{3} + 2x^{2} â€“ 9x â€“ 18 = (x â€“ 3) (x^{2} + 5x + 6)

= (x â€“ 3) (x + 3) (x + 2)

â‡’ The zeroes of x^{3} + 2x^{2} â€“ 9x â€“ 18 are 3, â€“3 and â€“ 2**Ques 40: If a and b are zeroes of the quadratic polynomial x ^{2} â€“ 6x + a; find the value of â€˜aâ€™ if 3Î± + 2Î² = 20.Sol: **We have quadratic polynomial = x

âˆµ a and b are zeroes of (1)

âˆ´

It is given that: 3Î± + 2Î² = 20 ...(2)

Now, Î± +Î² = 6 â‡’ 2 (Î±+ Î²) = 2(6)

2Î± + 2Î² = 12 ...(3)

Subtracting (3) from (2), we have

Substituting a = 8 in Î± + Î²= 6, we get

8 +Î² = 6 â‡’ Î² = â€“2

Since, Î±Î² = a

8(â€“2) = Î± â‡’ Î± = â€“16

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