Advertisements
Advertisements
Question
Find the quadratic polynomial, sum of whose zeroes is `( 5/2 )` and their product is 1. Hence, find the zeroes of the polynomial.
Advertisements
Solution
Let 𝛼 and 𝛽 be the zeroes of the required polynomial f(x).
Then (𝛼 + 𝛽) =` 5/2 `and 𝛼𝛽 = 1
`∴ f(x)=x^2-(∝+β) x+∝β `
`⇒f(x)=x^2-5/2x+1`
`⇒ f(x)=2x^2-5x+2`
Hence, the required polynomial is `f(x)=2x^2-5x+2`
∴` f(x) = 0 ⇒ 2x^2 – 5x + 2 = 0`
⇒ `2x^2 – (4x + x) + 2 = 0`
⇒` 2x^2 – 4x – x + 2 = 0`
⇒ `2x (x – 2) – 1 (x – 2) = 0`
⇒` (2x – 1) (x – 2) = 0`
⇒ `(2x – 1) = 0 or (x – 2) = 0`
`⇒ x =1/2 or x=2`
So, the zeros of f(x) are `1/2` and 2
APPEARS IN
RELATED QUESTIONS
Prove relation between the zeros and the coefficient of the quadratic polynomial ax2 + bx + c
Verify that the numbers given along side of the cubic polynomials are their zeroes. Also verify the relationship between the zeroes and the coefficients.
`2x^3 + x^2 – 5x + 2 ; 1/2, 1, – 2`
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
4s2 – 4s + 1
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
3x2 – x – 4
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 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 − 3x − 2, find a quadratic polynomial whose zeroes are `1/(2alpha+beta)+1/(2beta+alpha)`
If If α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3, find a polynomial whose roots are `(alpha-1)/(alpha+1)` , `(beta-1)/(beta+1)`
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) = (x^3 – 2x2 – 5x + 6)` and verify the relation between it zeros and coefficients.
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 are the zeroes of the polynomial `f(x) = 5x^2 -7x + 1` then `1/∝+1/β=?`
If 𝛼, 𝛽 are the zeroes of the polynomial f(x) = x2 + x – 2, then `(∝/β-∝/β)`
If x + 2 is a factor of x2 + ax + 2b and a + b = 4, then
Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial ______.
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:
`4x^2 + 5sqrt(2)x - 3`
If α and β are the zeros of a polynomial f(x) = px2 – 2x + 3p and α + β = αβ, then p is ______.
Find the zeroes of the quadratic polynomial 4s2 – 4s + 1 and verify the relationship between the zeroes and the coefficients.
