Advertisements
Advertisements
प्रश्न
Verify that 5, -2 and 13 are the zeroes of the cubic polynomial `p(x) = (3x^3 – 10x^2 – 27x + 10)` and verify the relation between its zeroes and coefficients.
उत्तर
p(x) = `(3x^3 – 10x^2 – 27x + 10)`
`p(5) = (3 × 5^3 – 10 × 5^2 – 27 × 5 + 10) = (375 – 250 – 135 + 10) = 0`
`p(–2) = [3 × (–2^3) – 10 × (–2^2) – 27 × (–2) + 10] = (–24 – 40 + 54 + 10) = 0 `
`p(1/3)={3xx(1/3)^3-10(1/3)^2-27xx1/3+10}=(3xx1/27-10xx1/9-9+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` ={5xx(-2)xx1/3}=-10/3=-("Constant term")/(("Coefficient of "x^3))`
APPEARS IN
संबंधित प्रश्न
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
3x2 – x – 4
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
`1/4 , -1`
If α and β are the zeroes of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/(aalpha+b)+1/(abeta+b)`.
If α and β are the zeros of the quadratic polynomial f(x) = x2 − p (x + 1) — c, show that (α + 1)(β +1) = 1− c.
Find the condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in A.P.
Find the quadratic polynomial, sum of whose zeroes is `sqrt2` and their product is `(1/3)`.
Find a cubic polynomial whose zeroes are 2, -3and 4.
If α, β, γ are the zeros of the polynomial f(x) = ax3 + bx2 + cx + d, then α2 + β2 + γ2 =
The product of the zeros of x3 + 4x2 + x − 6 is
If α and β are the zeros of a polynomial f(x) = px2 – 2x + 3p and α + β = αβ, then p is ______.
