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^{2}  4  8x
= 5x^{2}  8x  4
= 5x^{2}  10x + 2x  4
= 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^{2}  (Sum of the zeroes) x + (Product of the zeroes)
∴ The required polynomial is
= x^{2}  [8] x + [12]
= x^{2}  8x + 12
To find zeroes:
∵ x^{2}  8x + 12 = x^{2}  6x  2x + 12
= x (x  6)  2 (x  6)
= (x  6) (x  2)
∴ The zeroes of p (x) are 6 and 2.
Ques 23: If one zero of the polynomial (a^{2}  9) x^{2} + 13x + 6a is reciprocal of the other, find the value of ‘a’.
Sol: Here, p (x) = (a^{2}  9) x^{2} + 13x + 6a
Comparing it with Ax^{2} + Bx + C, we have:
A = (a^{2}  9); B = 13; C = 6
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’
Sol: Here, p (x) = ax^{2}  6x  6
∵ 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.
Sol: Here p (x) = x^{4} + x^{3}  34x^{2}  4x + 120
∵ 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^{2}  4 is a factor of p (x).
Now, dividing p(x) by x^{2}  4, we have:
∵ 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.
Sol: P(x) = 2x^{4} + 7x^{3}  19x^{2}  14x + 30
∵ √2 and √2 are the two zeroes of p (x).
∴ are the factors of p (x).
⇒ i.e., x^{2}  2 is a factor of p (x).
Now, dividing p (x) by x^{2}  2, we have:
∴ 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^{2}  (sum of the zeroes) x + (product of the zeroes)
∴ The required polynomial is
x^{2 } ( 2) x + ( 3)
⇒ x^{2} + 2x  3
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.
Sol: Here, p (x) = 4x^{2}  4x  3 = 4x^{2}  6x + 2x  3
= 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.
Sol: p (x) = 2x^{3 } 4x  x^{2} + 2
∵ √2 and √2 are the zeroes of p (x)
∴ and are the factors of p (x)
⇒ is a factor of p (x)
⇒ x^{2}  2 is a factor of p (x)
Now, Dividing p (x) by (x^{2}  2), we have:
⇒ 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.
Sol: p (x) = x^{4 } 3x^{3} + 6x  4
∵ √2 and ( √2) are the zeroes of p(x)
∴ x√2 and x( √2) are factors of p (x)
⇒ is a factor of p (x).
⇒ x^{2}  2 is a factor of p (x)
On Dividing p (x) by x^{2 } 2, we have:
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^{2}  [Sum of the zeroes] x + [Product of the zeroes] ...(1)
∵ 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^{2}  ( 1) x + ( 12)
= x^{2} + x  12 ...(2)
Comparing (2) with ax^{2} + bx + c, we have
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.
Ques 32: Using division algorithm, find the quotient and remainder on dividing f (x) by g (x), where f (x) = 6x^{3} + 13x^{2} + x  2 and g (x) = 2x + 1
Sol: Here, f (x) = 6x^{3} + 13x^{2} + x  2
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’.
Sol: We have:
∴ 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.
Sol: We have:
∴ 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.
Sol: p (x) = x^{3} + 3x^{2}  2x  6
∵ Two of its zeroes are √2 and √2
⇒ is a factor of p (x)
⇒ x^{2}  2 is a factor of p (x).
Now, dividing p (x) by x^{2}  2 we have:
∴ 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.
Sol: p (x) = 2x^{3} + x^{2}  6x  3
Two of its zeroes are √3 and √3
∴ and are factors of p (x)
i.e., is a factor of p (x)
⇒ x^{2}  3 is a factor of p (x)
Now, Dividing p (x) by x^{2}  3, we have:
∴ 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,
coefficient of x^{2} = 1
coefficient 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.
Sol: Let x^{3} + 2x^{2} + kx + 3 = p(x)
∵ The divisor = x – 3
∴ p(3) = 3^{3} + 2 × 3^{2} + 3k + 3
21 = 27 + 18 + 3k + 3
[∵ Remainder = 21]
⇒ 21 – 18 – 3 – 27 = 3k
⇒ –27 = 3k ⇒ k = – 9
Now, the given cubic polynomial
= x^{3} + 2x^{2} – 9x + 3
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^{2} – 6x + a ...(1)
∵ 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
116 videos420 docs77 tests

1. What is a polynomial? 
2. How do you identify the degree of a polynomial? 
3. Can a polynomial have negative exponents? 
4. How do you add or subtract polynomials? 
5. Can a polynomial have more than one variable? 

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