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**Q 1. Find the sum and product of zeroes of 3x ^{2} - 5x + 6.**

Here, p (x) = 3x

^{2}- 5x + 6

Comparing it with ax^{2}+ bx + c, we have

a = 3, b = - 5, c = 6

∴ Sum of the zeroes =and, Product of the zeroes =

**Q 2. Find the sum and product of the zeroes of polynomial p (x) = 2x ^{3} - 5x^{2} - 14x + 8.**

Comparing p (x) = 2x

^{3}- 5x^{2}- 14x + 8 with ax^{3}+ bx^{2}+ cx + d, we have

a = 2, b = –5,

c = - 14 and d = 8

∴ Sum of the zeroes =

Product of zeroes

**Q 3. Find a Quadratic polynomial whose zeroes are ****.**

Sum of zeroes (S)

Product of roots (P)

Since the required Quadratic polynomial

= k(x^{2}- Sx + P) ; where k is any real number.= k

Thus, the required polynomial is

= k (x^{2}- 2x - 1/4)

**Q 4. If α and β are the zeroes of a Quadratic polynomial x ^{2} + x - 2 then find the value of **

Comparing x

^{2}+ x - 2 with ax^{2}+ bx + c, we have:

a = 1, b = 1, c = - 2

Thus,

**Q 5. If a and b are the zeroes of x ^{2} + px + q then find the value of **

Comparing x

^{2}+ px + q with ax^{2}+ bx + c

a =1, b = p and c = q

∴ Sum of zeroes, a + b = - b/a

⇒

and αβ = c/a

⇒ αβ = q/1 = q

Now,Thus, the value of is

**Q 6. Find the zeroes of the quadratic polynomial 6x ^{2} - 3 - 7x.**

We have,

= 6x^{2}- 3 - 7x = 6x^{2}- 7x - 3

= 6x^{2}- 9x + 2x - 3

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

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

For 6x^{2}- 3 - 7x to be equal to zero,

either (3x + 1) = 0 or (2x - 3) = 0

⇒ 3x = - 1 or 2x = 3

⇒

Thus, the zeroes of and 3/2.

**Q 7. Find the zeroes of 2x ^{2} - 8x + 6.**

We have,

2x^{2}- 8x + 6 = 2x^{2}- 6x - 2x + 6

= 2x (x - 3) - 2 (x - 3)

= (2x - 2) (x - 3)

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

For 2x^{2}- 8x + 6 to be zero,

Either, x - 1 = 0 ⇒ x = 1

or x - 3 = 0 ⇒ x = 3

∴ The zeroes of 2x^{2}- 8x + 6 are 1 and 3.

**Q 8. Find the zeroes of the quadratic polynomial 3x ^{2} + 5x - 2.**

We have,

p (x) = 3x^{2}+ 5x - 2

= 3x^{2}+ 6x - x - 2

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

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

For p (x) = 0, we get

Either x + 2 = 0 ⇒ x = - 2

or 3x - 1 = 0 ⇒ x = 1/3

Thus, the zeroes of 3x^{2}+ 5x - 2 are - 2 and 1/3.

**Q 9. If the zero of a polynomial p (x) = 3x ^{2} - px + 2 and g (x) = 4x^{2} - q x - 10 is 2, then find the value of p and q.**

∵ p (x) = 3x

^{2}- px + 2

∴ p (2) = 3 (2)^{2}- p (2) + 2 = 0

[2 is a zero of p (x)]

or 12 - 2p + 2 or 14 - 2p = 0

or p = 7

Next g (x) = 4x^{2}- q x - 10

∴ g (2) = 4(2)^{2}- Q (2) - 10 = 0

[2 is a zero of g (x)]

or 4 × 4 - 2q - 10 = 0

or 16 - 2q - 10 = 0

or 6 - 2q = 0

⇒ q = 6/2 ⇒ q = 3

Thus, the required values are p = 7 and q = 3.

**Q 10. Find the value of ‘k’ such that the quadratic polynomial 3x ^{2} + 2kx + x - k - 5 has the sum of zeroes as half of their product.**

Here, p (x) = 3x

^{2}+ 2kx + x - k - 5

= 3x^{2}+ (2k + 1) x - (k + 5)

Comparing p (x) with ax^{2}+ bx + c, we have:

a = 3, b = (2k + 1),

c = - (k + 5)

∴ Sum of the zeroes

Product of the zeroes

According to the condition,

Sum of zeroes = 1/2 (product of roots)

⇒ - 2 (2k + 1) = - (k + 5)

⇒ 2 (2k + 1) = k + 5

⇒ 4k + 2 = k + 5

⇒ 4k - k = 5 - 2

⇒ 3k = 3

⇒ k = 3/3 = 1

**Q 11. On dividing p (x) by a polynomial x - 1 - x ^{2}, the Quotient and remainder were (x - 2) and 3 respectively. Find p (x).**

Here,dividend = p (x)

Divisor, g (x) = (x - 1 - x^{2})

Quotient, q(x) = (x - 2)

Remainder, r (x) = 3

∵ Dividend = [Divisor × Quotient] +Remainder

∴ p (x)= [g (x) × q(x)] + r (x)

= [(x - 1 - x^{2}) (x - 2)] + 3

= [x^{2}- x - x^{3}- 2x + 2 + 2x^{2}] + 3

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

= - x^{3}+ 3x^{2 }- 3x + 5

**Q 12. Find the zeroes of the polynomial f (x) = 2 - x ^{2}.**

We have f (x)= 2 - x

^{2}

= (√2 )^{2 }- x^{2}

**Q 13. Find the cubic polynomial whose zeroes are 5, 3 and - 2.**

∵ 5, 3 and - 2 are zeroes of p (x)

∴ (x - 5), (x - 3) and (x + 2) are the factors of p (x)

⇒ p (x) = k (x - 5) (x - 3) (x + 2)

= k (x^{2}- 8x + 15) (x + 2)

= k (x^{3}- 8x^{2}+ 15x + 2x^{2}- 16x + 30

= k (x^{3}+ [- 8 + 2] x^{2}+ [15 - 16] x + 30)

= k (x^{3 }- 6x^{2}- x + 30)

Thus, the required polynomial is k (x^{3}- 6x^{2}- x + 30).

**Q 14. If α, β and γ be the zeroes of a polynomial p (x) such that (α + β +γ) = 3, (αβ + βγ + γα) = -10 and αβγ = - 24 then find p (x).**

Here, α + β + γ = 3

αb + βγ + γα = - 10

αβγ = - 24

∵ A cubic polynomial having zeroes as α,β,γ is

p (x) = x^{3}- (a + b + γ) x^{2 }+ (αβ + βγ + γα) x - (αβγ)

∴The required cubic polynomial is

= k {x^{3}- (3) x^{2}+ (- 10) x - (- 24)}

= k(x^{3 }- 3x^{2}- 10x + 24)

Note:If α, β and γ be the zeroes of a cubic polynomial p (x) then

p (x) = x^{3}- [Sum of the zeroes] x^{2}+ [Product of the zeroes taken two at a time] x - [Product of zeroes]

i.e., p (x) = k {x^{3}- (α + β + γ) x^{2}+ [αβ + βγ + γα] x - (αβγ).

**Q 15. Find all the zeroes of the polynomial 4x ^{4} - 20x^{3} + 23x^{2} + 5x - 6 if two of its zeroes are 2 and 3.**

Here, p (x) =4x

^{4}- 20x^{3}+ 23x^{2}+ 5x - 6

Since, 2 and 3 are the zeroes of p (x),

∴ (x - 2) and (x - 3) are the factors of p(x)

⇒ (x - 2) (x - 3) is a factor of p (x)

⇒ x^{2 }- 5x + 6 is a factor of p (x)

Now, using the division algorithm for x^{2}- 5x + 6 and the given polynomial p (x), we

get:∴ We get (x

^{2}- 5x + 6) (4x^{2}- 1) = p (x)

⇒ (x - 3) (x - 2) [(2x)^{2}- (1)2] = p (x)

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

⇒

Thus, all the zeroes of p (x) are:

**Q 16. If 1 is a zero of x ^{3} - 3x^{2} - x + 3 then find all other zeroes.**

Here,p (x) = x

^{3}- 3x^{2}- x + 3

∵ 1 is a zero of p (x)

∴ (x - 1) is a factor of p (x).

Now, dividing p (x) by (x - 1), we have:∴ p (x) = (x - 1) (x

^{2}- 2x - 3)

⇒ p (x) = (x - 1) [(x^{2}- 3x + x - 3)]

= (x - 1) [x (x - 3) + 1 (x - 3)]

= (x - 1) [(x - 3) (x + 1)]

i.e., (x - 1), (x - 3) and (x + 1) are the factors of p (x).

⇒ 1, 3, and - 1 are the zeroes of p (x).

**Q 17. Find all the zeroes of 2x ^{4} - 3x^{3} - 3x^{2} + 6x - 2, if two of its zeroes are 1 and 1/2.**

Here, p(x) = 2x

^{4}- 3x^{3}- 3x^{2}+ 6x - 2

∵ 1 and are the zeroes of p (x)

∴ (x - 1) and are the factors of p (x)

⇒ (x - 1) (2x - 1) is a factor of p (x)

⇒ (2x^{2}- 3x + 1) is a factor of p (x).

Now, dividing p (x) by 2x^{2}- 3x + 1, we get∴ p (x) = (2x

^{2}− 3x + 1) (x^{2}− 2)

are the zeroes of p (x).

**Q 18. On dividing 4x ^{3} - 8x^{2} + 8x + 1 by a polynomial g (x), the Quotient and remainder were (2x^{2} - 3x + 2) and (x + 3) respectively. Find g (x).**

∵ Dividend = Divisor × Quotient + Remainder

i.e., p (x) = g (x) × Q (x) + r (x)

∴ g (x) =Thus, the required polynomial g (x) = 2x - 1.

**Q 19. If α and β are the zeroes of the quadratic polynomial p (x) = kx ^{2} + 4x + 4 such that α^{2} + β^{2} = 24, find the value of k.**

Here, p (x) = kx

^{2}+ 4x + 4.

Comparing it with ax^{2}+ bx + c, we have:

a = k; b = 4; c = 4

∴ Sum of the zeroes = -b/a

⇒ α + β = -4/k

and Product of the zeroes = c/a

⇒ αβ = 4/k

∵ α^{2}+ β^{2 }= 24

∴ (α + β)^{2}- 2αβ = 24

[∵ (x + y)^{2}= x^{2}+ y^{2}+ 2xy ⇒ (x + y)^{2}- 2xy = x^{2}+ y^{2}]

⇒

⇒

⇒ 16 − 8k − 24k^{2}=0

⇒ 24k^{2}+ 8k − 16 = 0

⇒ (3k − 2) (k + 1) = 0

⇒ 3k − 2= 0 or k + 1 = 0

⇒ k = 2/3 or k = -1

**Q 20. Find the zeroes of the quadratic polynomial 6x ^{2} - 3 - 7x and verify the relationship between the zeroes and the coefficients of the polynomial.**

Here, p (x) = 6x

^{2}- 3 - 7x = 6x^{2}- 7x - 3

= 6x^{2 }- 9x + 2x - 3

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

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

=

∴ Zeroes of p (x) are 3/2 and

To verify the relationship:

Sum of the zeroes =⇒

⇒

⇒ 7/6 = 7/6

L.H.S = R.H.S ⇒ Relationship is verified.

Product of the zeroes =⇒

⇒

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

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