Advertisements
Advertisements
प्रश्न
Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.
Advertisements
उत्तर
Given that a, a + b, a + 2b are roots of given polynomial x3 – 6x2 + 3x + 10
Sum of the roots
⇒ a + 2b + a + a + b = `(-("coefficient of" x^2))/("coefficient of" x^3)`
⇒ 3a + 3b = `(-(-6))/1` = 6
⇒ 3(a + b) = 6
⇒ a + b = 2 .........(1)
b = 2 – a
Product of roots
⇒ (a + 2b)(a + b)a = `(-("constant"))/("coefficient of" x^3)`
⇒ (a + b + b)(a + b)a = `-10/1`
Substituting the value of a + b = 2 in it
⇒ (2 + b)(2)a = –10
⇒ (2 + b)2a = –10
⇒ (2 + 2 – a)2a = –10
⇒ (4 – a)2a = –10
⇒ 4a – a2 = –5
⇒ a2 – 4a – 5 = 0
⇒ a2 – 5a + a – 5 = 0
⇒ (a – 5)(a + 1) = 0
a – 5 = 0 or a + 1 = 0
a = 5 a = –1
a = 5, –1 in (1) a + b = 2
When a = 5,
5 + b = 2
⇒ b = –3
a = –1, –1 + b = 2
⇒ b = 3
∴ If a = 5 then b = –3
or
If a = –1 then b = 3
APPEARS IN
संबंधित प्रश्न
Find the zeros of the quadratic polynomial 6x2 - 13x + 6 and verify the relation between the zero and its coefficients.
Verify that the numbers given along side of the cubic polynomials are their zeroes. Also, verify the relationship between the zeros and the coefficients.
`f(x) = 2x^3 + x^2 - 5x + 2 ; 1/2, 1, - 2`
Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients:
`q(x) = sqrt(3)x^2 + 10x + 7sqrt(3)`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α4 + β4
If α and β are the zeros of the quadratic polynomial p(s) = 3s2 − 6s + 4, find the value of `alpha/beta+beta/alpha+2[1/alpha+1/beta]+3alphabeta`
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`.
If α and β are the zeros of a quadratic polynomial such that α + β = 24 and α – β = 8, find a quadratic polynomial having α and β as its zeros.
If α and β are the zeroes of the polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α − β)2.
If the zeros of the polynomial f(x) = x3 – 12x2 + 39x + k are in A.P., find the value of k.
Find the zeroes of the quadratic polynomial` (x^2 ˗ 5)` and verify the relation between the zeroes and the coefficients.
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
By actual division, show that x2 – 3 is a factor of 2x4 + 3x3 – 2x2 – 9x – 12.
If 3 and –3 are two zeroes of the polynomial (x4 + x3 – 11x2 – 9x + 18), find all the zeroes of the given polynomial.
If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]
Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is ______.
For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
`-2sqrt(3), -9`
A quadratic polynomial the sum and product of whose zeroes are – 3 and 2 respectively, is ______.
Find the zeroes of the quadratic polynomial x2 + 6x + 8 and verify the relationship between the zeroes and the coefficients.
If α, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of α–1 + β–1.
