Advertisements
Advertisements
प्रश्न
If the zeroes of the polynomial x3 – 3x2 + x + 1 are a – b, a, a + b, find a and b
Advertisements
उत्तर १
Since, (a - b), a, (a + b) are the zeroes of the polynomial x3 – 3x2 + x + 1.
Therefore, sum of the zeroes = (a - b) + a + (a + b) = -(-3)/1 = 3
⇒ 3a = 3 ⇒ a =1
∴ Sum of the products of is zeroes taken two at a time = a(a - b) + a(a + b) + (a + b) (a - b) =1/1 = 1
a2 - ab + a2 + ab + a2 - b2 = 1
⇒ 3a2 - b2 =1
Putting the value of a,
⇒ 3(1)2 - b2 = 1
⇒ 3 - b2 = 1
⇒ b2 = 2
⇒ b = ±`sqrt2`
Hence, a = 1 and b = ± `sqrt2`
उत्तर २
p(x) = x3 - 3x2 + x + 1
Zeroes are a - b, a + a + b
Comparing the given polynomial with px3 + qx2 + rx + 1, we obtain
p = 1, q = - 3, r = 1, t = 1
Sum of zeroes = a - b + a + a +b
`(-q)/p = 3a`
`(-(-3))/1 =3a`
3 = 3a
a = 1
The zeroes are 1-b, 1, 1+b
Multiplication of zeroes = 1(1-b)(1+b)
`(-t)/p=1-b^2`
`(-1)/1 = 1 - b^2`
1- b2 = -1
1 + 1 = b2
b = ±`sqrt2`
Hence a = 1 and b = `sqrt2` or - `sqrt2`
APPEARS IN
संबंधित प्रश्न
Prove relation between the zeros and the coefficient of the quadratic polynomial ax2 + bx + c
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
t2 – 15
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
`1/4 , -1`
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
4, 1
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α - β
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha+1/beta-2alphabeta`
If 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x2 − 5x −1, find the value of α2β + αβ2.
If α and β are the zeros of the quadratic polynomial f(x) = x2 − px + q, prove that `alpha^2/beta^2+beta^2/alpha^2=p^4/q^2-(4p^2)/q+2`
Verify that 3, -2, 1 are the zeros of the cubic polynomial `p(x) = (x^3 – 2x2 – 5x + 6)` and verify the relation between it zeros and coefficients.
If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]
If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are each equal to zero, then the third zero is
If one of the zeroes of the quadratic polynomial (k – 1)x2 + k x + 1 is –3, then the value of k is ______.
Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is ______.
The zeroes of the quadratic polynomial x2 + 99x + 127 are ______.
If α, β are the zeroes of the polynomial p(x) = 4x2 – 3x – 7, then `(1/α + 1/β)` is equal to ______.
Find the zeroes of the quadratic polynomial x2 + 6x + 8 and verify the relationship between the zeroes and the coefficients.
Find the zeroes of the polynomial x2 + 4x – 12.
If α, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of α2 + β2.
Find the zeroes of the quadratic polynomial 4s2 – 4s + 1 and verify the relationship between the zeroes and the coefficients.
