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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)

∴ zeroes of p (x) are 2 and

Relationship Verification

Sum of the zeroes

⇒

⇒

⇒ 8/5 = 8/5

i.e., L.H.S. = R.H.S. ⇒ relationship is verified.

Product of the zeroes =

⇒

⇒

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

⇒ 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:** ∵ The quadratic polynomial p (x) is given by

x

∴ The required polynomial is

= x

= x

To find zeroes:

∵ x

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

= (x - 6) (x - 2)

∴ The zeroes of p (x) are 6 and 2.

Sol:

Comparing it with Ax

A = (a

Let one of the zeroes = a

∴ The other zero = 1/α

Now, Product of the zeroes

⇒

⇒ 6a = a^{2} − 9 ⇒ a^{2} − 6a + 9 = 0

⇒ (a − 3)^{2} =0 ⇒ a − 3=0

⇒ 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 ‘a’**

∵ Product of zeroes =

but product of zeroes is given as 4

∴ ⇒ − 6 = 4 × a

⇒ ⇒

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:

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

∴ (x - 2) and (x + 2) are factors of p (x)

⇒ (x - 2) (x + 2) is a factor of p (x)

⇒ x

Now, dividing p(x) by x

∵ Remainder = 0

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

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

∵ 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).

⇒ 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 √2 and -√2. **

∵ √2

∴ are the factors of p (x).

⇒ i.e., x

Now, dividing p (x) by x

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

[∵ Remainder = 0]

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

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

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

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

=

∴ 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: **∵ The given zeroes are 1 and - 3.

∴ Sum of the zeroes = 1 + (- 3) = - 2

Product of the zeroes = 1 × (- 3) = - 3

A quadratic polynomial p (x) is given by

x

∴ The required polynomial is

x

⇒ x

Verification of relationship

∵ Sum of the zeroes

∴

⇒− 2= − 2

i.e., L.H.S = R.H.S ⇒ The sum of zeroes is verified

∵ Product of the zeroes =

∴

⇒− 3= − 3

i.e., L.H.S = R.H.S ⇒ 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)

=

∴ are zeroes of p (x).

Verification of relationship

∵ Sum of the zeroes =

∴

⇒ 2/2 = 1 ⇒ 1= 1

⇒

2/2 = 1 ⇒ 1 = 1

i.e., L.H.S = R.H.S ⇒ Sum of zeroes is verified

Now, Product of zeroes =

⇒

i.e., L.H.S = R.H.S ⇒ 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 √2 and -√2.**

∵

∴ and are the factors of p (x)

⇒ is a factor of p (x)

⇒ x

Now, Dividing p (x) by (x

⇒ p (x) = (x^{2} - 2) (2x - 1)

∴ 2x - 1 is also a factor of p (x)

i.e.,is another factor of px.

⇒ 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 √2 and - √2.**p (x) = x

Sol:

∵

∴ x-

⇒ is a factor of p (x).

⇒ x

On Dividing p (x) by x

Since, remainder = 0

∴ (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)

∴ 1 and 2 are zeroes of p (x).

∴ All the zeroes of p (x) are ,**√**2 , **- √**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

∵ The given zeroes are - 4 and 3

∴ Sum of the zeroes = (- 4) + 3 = - 1

Product of the zeroes = (- 4) × 3 = - 12

From (1), we have

x

= x

Comparing (2) with ax

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

∴ Sum of the zeroes = -b/a

⇒ (+ 3) + (- 4) = -1/1

i.e., L.H.S = R.H.S ⇒ Sum of zeroes is verified.

Product of zeroes = c/a

⇒ 3 × (- 4) = -12/1

⇒ - 12 = - 12

i.e., L.H.S = R.H.S ⇒ 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 ‘a’ and ‘b’.**We have:

Sol:

∴ 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:

∴ 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 - √2 and √2.**p (x) = x

Sol:

∵ Two of its zeroes are -√2 and √2

⇒ is a factor of p (x)

⇒ x

Now, dividing p (x) by x

∴ p (x) = (x^{2} - 2) (x + 3)

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

⇒ (- 3) is a zero of p (x)

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

Sol:

Two of its zeroes are -√3

∴ and are factors of p (x)

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

⇒ x

Now, Dividing p (x) by x

∴ p (x) = (x^{2} - 3) (2x + 1)

⇒ is a factor of p (x)

⇒ -1/2 is a zero of p (x)

∴ All the zeroes of p (x) are -√3 , √3 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

∴ zeroes of the given polynomial are

Now in,

co-efficient of x^{2} = 1

co-efficient of x = 1/6

constant term = –2

∴ 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 = √2

Product of zeroes

∵ A quadratic polynomial is given by

x^{2} – [sum of roots] x + [Product of roots]

∴ The required polynomial is

⇒

Since =

⇒ 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.**

∵ The divisor = x – 3

∴ p(3) = 3

21 = 27 + 18 + 3k + 3

[∵ Remainder = 21]

⇒ 21 – 18 – 3 – 27 = 3k

⇒ –27 = 3k ⇒ k = – 9

Now, the given cubic polynomial

= x

since,

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

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

= (x – 3) (x + 3) (x + 2)

⇒ The zeroes of x^{3} + 2x^{2} – 9x – 18 are 3, –3 and – 2**Ques 40: If a and b are zeroes of the quadratic polynomial x ^{2} – 6x + a; find the value of ‘a’ if 3α + 2β = 20.Sol: **We have quadratic polynomial = x

∵ a and b are zeroes of (1)

∴

It is given that: 3α + 2β = 20 ...(2)

Now, α +β = 6 ⇒ 2 (α+ β) = 2(6)

2α + 2β = 12 ...(3)

Subtracting (3) from (2), we have

Substituting a = 8 in α + β= 6, we get

8 +β = 6 ⇒ β = –2

Since, αβ = a

8(–2) = α ⇒ α = –16

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