Advertisements
Advertisements
Question
Verify that 3, -2, 1 are the zeros of the cubic polynomial `p(x) = (x^3 – 2x2 – 5x + 6)` and verify the relation between it zeros and coefficients.
Advertisements
Solution
The given polynomial is `p(x) = (x^3 – 2x^2 – 5x + 6)`
`∴ p(3) = (3^3 – 2 × 3^2 – 5 × 3 + 6) = (27 – 18 – 15 + 6) = 0`
`p(-2) = [ (– 2^3) – 2 × (– 2)^2 – 5 × (– 2) + 6] = (–8 –8 + 10 + 6) = 0`
`p(1) = (1^3 – 2 × 1^2 – 5 × 1 + 6) = ( 1 – 2 – 5 + 6) = 0`
∴ 3, –2 and 1are the zeroes of p(x),
Let 𝛼 = 3, 𝛽 = –2 and γ = 1. Then we have:
(𝛼 + 𝛽 + γ) = (3 – 2 + 1) = 2 = `(-("Coefficient of" x^2))/(("Coefficient of" x^2))`
(𝛼𝛽 + 𝛽γ + γ𝛼) = (–6 –2 + 3) = `−5/1 = ("Coefficient of x")/("Coefficient of x"^2)`
𝛼𝛽γ = { 3 × (-2) × 1}=`(-6)/1= -(("Constant term"))/(("Coefficient of" x^3))`
APPEARS IN
RELATED QUESTIONS
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes, respectively.
`sqrt2 , 1/3`
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`0, sqrt5`
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`-1/4 ,1/4`
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case
x3 – 4x2 + 5x – 2; 2, 1, 1
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`q(x)=sqrt3x^2+10x+7sqrt3`
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`g(x)=a(x^2+1)-x(a^2+1)`
If 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x2 − 5x −1, find the value of α2β + αβ2.
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 sum of the zeros of the quadratic polynomial f(t) = kt2 + 2t + 3k is equal to their product, find the value of k.
On dividing `3x^3 + x^2 + 2x + 5` is divided by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).
Find all the zeroes of `(x^4 + x^3 – 23x^2 – 3x + 60)`, if it is given that two of its zeroes are `sqrt3 and –sqrt3`.
If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]
What should be subtracted to the polynomial x2 − 16x + 30, so that 15 is the zero of the resulting polynomial?
A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, 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.
`21/8, 5/16`
The only value of k for which the quadratic polynomial kx2 + x + k has equal zeros is `1/2`
Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x and verify the relationship between the zeroes and the coefficients.
A quadratic polynomial the sum and product of whose zeroes are – 3 and 2 respectively, is ______.
Find the zeroes of the quadratic polynomial 4s2 – 4s + 1 and verify the relationship between the zeroes and the coefficients.
