Advertisements
Advertisements
Question
If 2 and -2 are two zeroes of the polynomial `(x^4 + x^3 – 34x^2 – 4x + 120)`, find all the zeroes of the given polynomial.
Advertisements
Solution
Let f(x) = x^4 + x^3 – 34x^2 – 4x + 120
Since 2 and -2 are the zeroes of f(x), it follows that each one of (x – 2) and (x + 2) is a factor of f(x).
Consequently, (x – 2) (x + 2) = (x2 – 4) is a factor of f(x).
On dividing f(x) by `(x^2 – 4)`, we get:
f(x) = 0
⇒` (x2 + x – 30) (x^2 – 4) = 0`
⇒` (x^2 + 6x – 5x – 30) (x – 2) (x + 2)`
⇒ `[x(x + 6) – 5(x + 6)] (x – 2) (x + 2)`
⇒`(x – 5) (x + 6) (x – 2) (x + 2) = 0`
⇒` x = 5 or x = -6 or x = 2 or x = -2`
Hence, all the zeroes are 2, -2, 5 and -6.
APPEARS IN
RELATED QUESTIONS
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α4 + β4
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(t) = t2 − 4t + 3, find the value of α4β3 + α3β4.
If α and β are the zeros of the quadratic polynomial f(x) = x2 − 1, find a quadratic polynomial whose zeroes are `(2alpha)/beta" and "(2beta)/alpha`
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 `2x^2 ˗ 11x + 15` 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.
Find a cubic polynomial whose zeroes are `1/2, 1 and -3.`
A quadratic polynomial, whose zeroes are –3 and 4, is ______.
Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1?
If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial 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:
3x2 + 4x – 4
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:
t3 – 2t2 – 15t
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`
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`
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`7y^2 - 11/3 y - 2/3`
If α and β are the zeros of a polynomial f(x) = px2 – 2x + 3p and α + β = αβ, then p is ______.
