Advertisements
Advertisements
प्रश्न
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
`1/4 , -1`
Advertisements
उत्तर
Given: α + β = `1/4`, αβ = -1
Since ax2 + bx + c = kx2 - k(α + β)x + kαβ
In comparison,
a = k, b = -k(α + β) and c = kαβ
α + β = `(-b)/a = 1/4` and αβ = `c/a = -1`
⇒ a = 4
⇒ b = -4(α + β)
⇒ c = kαβ = 4(-1)
Hence, on writing as ax2 + bx + c
⇒ 4x2 - 4(α + β)x + 4(αβ)
⇒ `4x^2 - 4(1/4)x + 4(-1)`
⇒ 4x2 - x - 4
The quadratic polynomial is 4x2 - x - 4.
APPEARS IN
संबंधित प्रश्न
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
4u2 + 8u
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 a and 3 are the zeros of the quadratic polynomial f(x) = x2 + x − 2, find the value of `1/alpha-1/beta`.
If α and β are the zeros of the quadratic polynomial p(y) = 5y2 − 7y + 1, find the value of `1/alpha+1/beta`
If 1 and –2 are two zeroes of the polynomial `(x^3 – 4x^2 – 7x + 10)`, find its third zero.
Define a polynomial with real coefficients.
What should be added to the polynomial x2 − 5x + 4, so that 3 is the zero of the resulting polynomial?
The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is
Case Study -1
The figure given alongside shows the path of a diver, when she takes a jump from the diving board. Clearly it is a parabola.
Annie was standing on a diving board, 48 feet above the water level. She took a dive into the pool. Her height (in feet) above the water level at any time ‘t’ in seconds is given by the polynomial h(t) such that h(t) = -16t2 + 8t + k.
The zeroes of the polynomial r(t) = -12t2 + (k - 3)t + 48 are negative of each other. Then k is ______.
A quadratic polynomial, whose zeroes are –3 and 4, is ______.
If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the product of the other two zeroes is ______.
Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1?
If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
5t2 + 12t + 7
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`4x^2 + 5sqrt(2)x - 3`
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.
`(-3)/(2sqrt(5)), -1/2`
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.
If p(x) = x2 + 5x + 6, then p(– 2) is ______.
If α, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of α2 + β2.
