Advertisements
Advertisements
Question
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
1, 1
Advertisements
Solution
Given: α + β = 1, αβ = 1
Since ax2 + bx + c = k[x2 - (α + β)x + αβ]
Or `(ax^2 + bx + c)/k = (x^2 - 1x + 1)`
Or `(ax^2 + bx + c)/k = (x^2 - x + 1)/1`
Here k is a constant term, by comparing k = 1
Hence, ax2 + bx + c = x2 - x + 1
The quadratic polynomial is x2 – x + 1.
APPEARS IN
RELATED QUESTIONS
Find all zeroes of the polynomial `(2x^4 - 9x^3 + 5x^2 + 3x - 1)` if two of its zeroes are `(2 + sqrt3)` and `(2 - sqrt3)`
Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients:
g(x) = a(x2 + 1) – x(a2 + 1)
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `beta/(aalpha + b) + alpha/(abeta + b)`.
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(t) = t2 − 4t + 3, find the value of α4β3 + α3β4.
If α and β are the zeros of the quadratic polynomial f(x) = x2 – 1, find a quadratic polynomial whose zeros are `(2alpha)/beta` and `(2beta)/alpha`.
Find the zeroes of the quadratic polynomial f(x) = 4x2 – 4x – 3 and verify the relation between its zeroes and coefficients.
Find the quadratic polynomial whose zeroes are `2/3` and `(-1)/4`. Verify the relation between the coefficients and the zeroes of the polynomial.
Find the quadratic polynomial, sum of whose zeroes is `sqrt2` and their product is `(1/3)`.
If (x+a) is a factor of the polynomial `2x^2 + 2ax + 5x + 10`, find the value of a.
If 1 and –2 are two zeroes of the polynomial (x3 – 4x2 – 7x + 10), find its third zero.
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\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]
A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is
An asana is a body posture, originally and still a general term for a sitting meditation pose, and later extended in hatha yoga and modern yoga as exercise, to any type of pose or position, adding reclining, standing, inverted, twisting, and balancing poses. In the figure, one can observe that poses can be related to representation of quadratic polynomial.
The zeroes of the quadratic polynomial `4sqrt3"x"^2 + 5"x" - 2sqrt3` are:
If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
If all three zeroes of a cubic polynomial x3 + ax2 – bx + c are positive, then at least one of a, b and c is non-negative.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
3x2 + 4x – 4
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 factorization.
`-8/3, 4/3`
Find the zeroes of the quadratic polynomial 4s2 – 4s + 1 and verify the relationship between the zeroes and the coefficients.
