Advertisements
Advertisements
प्रश्न
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
t3 – 2t2 – 15t
Advertisements
उत्तर
t3 – 2t2 – 15t
Taking t common, we get,
t(t2 – 2t – 15)
Splitting the middle term of the equation t2 – 2t – 15, we get,
t(t2 – 5t + 3t – 15)
Taking the common factors out, we get,
t(t(t – 5) + 3(t – 5)
On grouping, we get,
t(t + 3)(t – 5)
So, the zeroes are,
t = 0
t + 3 = 0
`\implies` t = – 3
t – 5 = 0
`\implies` t = 5
Therefore, zeroes are 0, 5 and – 3
Verification:
Sum of the zeroes = – (coefficient of x2) ÷ coefficient of x3
α + β + γ = `- b/a`
(0) + (– 3) + (5) = `- (-2)/1`
= 2
Sum of the products of two zeroes at a time = coefficient of x ÷ coefficient of x3
αβ + βγ + αγ = `c/a`
(0)(– 3) + (– 3)(5) + (0)(5) = `-15/1`
= – 15
Product of all the zeroes = – (constant term) ÷ coefficient of x3
αβγ = `- d/a`
(0)(– 3)(5) = 0
= 0
APPEARS IN
संबंधित प्रश्न
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
6x2 – 3 – 7x
If α and β are the zeros of the quadratic polynomial f(x) = 6x2 + x − 2, find the value of `alpha/beta+beta/alpha`.
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 one zero of the quadratic polynomial f(x) = 4x2 − 8kx − 9 is negative of the other, find the value of k.
If α and β are the zeros of the quadratic polynomial f(x) = x2 − 3x − 2, find a quadratic polynomial whose zeroes are `1/(2alpha+beta)+1/(2beta+alpha)`
If α and β are the zeroes of the polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α − β)2.
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 quadratic polynomial `4x^2 - 4x + 1` and verify the relation between the zeroes and the coefficients.
The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`2x^2 + (7/2)x + 3/4`
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.
The zeroes of the quadratic polynomial x2 + 99x + 127 are ______.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`y^2 + 3/2 sqrt(5)y - 5`
If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is reciprocal of the other, then find the value of k.
If α, β are the zeroes of the polynomial p(x) = 4x2 – 3x – 7, then `(1/α + 1/β)` is equal to ______.
A quadratic polynomial the sum and product of whose zeroes are – 3 and 2 respectively, is ______.
If p(x) = x2 + 5x + 6, then p(– 2) is ______.
A quadratic polynomial whose sum and product of zeroes are 2 and – 1 respectively is ______.
Find the zeroes of the polynomial x2 + 4x – 12.
The zeroes of the polynomial p(x) = 25x2 – 49 are ______.
