Advertisements
Advertisements
प्रश्न
Verify that 5, –2 and `1/3` are the zeroes of the cubic polynomial p(x) = (3x3 – 10x2 – 27x + 10) and verify the relation between its zeros and coefficients.
Advertisements
उत्तर
p(x) = (3x3 – 10x2 – 27x + 10)
p(5) = (3 × 53 – 10 × 52 – 27 × 5 + 10)
= (375 – 250 – 135 + 10)
= 0
p(–2) = [3 × (–23) – 10 × (–22) – 27 × (–2) + 10]
= (–24 – 40 + 54 + 10)
= 0
`p(1/3) = {3 xx (1/3)^3 - 10 xx (1/3)^2 - 27 xx 1/3 + 10}`
= `(3 xx 1/27 - 10 xx 1/9 - 9 + 10)`
= `(1/9 - 10/9 + 1)`
= `((1 - 10 - 9)/9)`
= `(0 / 9)`
= 0
∴ 5, –2 and `1/3` are the zeroes of p(x).
Let α = 5, β = –2 and γ = `1/3`.
Then we have:
`(α + β + γ) = (5 - 2 + 1/3)`
= `10/3`
= `(-("Coefficient of" x^2))/(("Coefficient of" x^3))`
`(αβ + βγ + γα) = (-10 - 2/3 + 5/3)`
= `(-27)/3`
= `("Coefficient of" x)/("Coefficient of"" x^3)`
`αβγ ={5 xx (-2) xx 1/3}`
= `(-10)/3`
= `(-("Constant term"))/(("Coefficient of" x^3))`
APPEARS IN
संबंधित प्रश्न
If 𝛼 and 𝛽 are the zeros of the quadratic polynomial f(x) = x2 − 5x + 4, find the value of `1/alpha+1/beta-2alphabeta`
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 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 the squared difference of the zeros of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the value of p.
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 f(x) = 4x2 – 4x – 3 and verify the relation between its zeroes and coefficients.
Find the zeroes of the polynomial f(x) = `2sqrt(3)x^2 - 5x + sqrt(3)` and verify the relation between its zeroes and coefficients.
Find the zeroes of the quadratic polynomial (3x2 – x – 4) and verify the relation between the zeroes and the coefficients.
If (x+a) is a factor of the polynomial `2x^2 + 2ax + 5x + 10`, find the value of a.
Verify that 3, –2, 1 are the zeros of the cubic polynomial p(x) = (x3 – 2x2 – 5x + 6) and verify the relation between it zeros and coefficients.
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 one of the zeroes of the quadratic polynomial (k – 1)x2 + k x + 1 is –3, then the value of k is ______.
If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
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`
Given that `sqrt(2)` is a zero of the cubic polynomial `6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2)`, find its other two zeroes.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`v^2 + 4sqrt(3)v - 15`
If the zeroes of the polynomial x2 + px + q are double in value to the zeroes of the polynomial 2x2 – 5x – 3, then find the values of p and q.
