Advertisements
Advertisements
Question
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)`
Advertisements
Solution
Since α and β are the zeros of the quadratic polynomial f(x) = x2 − 3x − 2
The roots are α and β
`alpha+beta="-coefficient of x"/("coefficient of "x^2)`
`alpha+beta=-((-3)/1)`
α + β = -(-3)
α + β = 3
`alphabeta="constant term"/("coefficient of "x^2)`
`alphabeta=(-2)/1`
αβ = -2
Let S and P denote respectively the sum and the product of zero of the required polynomial . Then,
`S=1/(2alpha+beta)+1/(2beta+alpha)`
Taking least common factor then we have ,
`S=1/(2alpha+beta)xx(2beta+alpha)/(2beta+alpha)+1/(2beta+alpha)xx(2alpha+beta)/(2alpha+beta)`
`S=(2beta+alpha)/((2alpha+beta)(2beta+alpha))+(2alpha+beta)/((2beta+alpha)(2alpha+beta))`
`S=(2beta+alpha+2alpha+beta)/((2alpha+beta)(2beta+alpha))`
`S=(3beta+3alpha)/(4alphabeta+2beta^2+2alpha^2+betaalpha)`
`S=(3(beta+alpha))/(5alphabeta+2(alpha^2+beta^2))`
`S=(3(beta+alpha))/(5alphabeta+2[(alpha+beta)^2-2alphabeta])`
By substituting α + β = 3 and αβ = -2 we get,
`S=(3(3))/(5(-2)+2[(3)^2-2xx-2])`
`S=9/(-10+2(13))`
`S=9/(-10+26)`
`S=9/16`
`P=1/(2alpha+beta)xx1/(2beta+alpha)`
`P=1/((2alpha+beta)(2beta+alpha))`
`P=1/(4alphabeta+2beta^2+2alpha^2+betaalpha)`
`P=1/(5alphabeta+2(alpha^2+beta^2))`
`P=1/(5alphabeta+2[(alpha+beta)^2-2alphabeta])`
By substituting α + β = 3 and αβ = -2 we get,
`P=1/(5(-2)+2[(3)^2-2xx-2])`
`P=1/(10+2[9+4])`
`P=1/(10+2(13))`
`P=1/(-10+26)`
`P=1/16`
Hence ,the required polynomial f(x) is given by
`f(x) = k(x^2 - Sx + P)`
`f(x) = k(x^2-9/16x+1/16)`
Hence, the required equation is `f(x) = k(x^2-9/16x+1/16)` Where k is any non zero real number.
APPEARS IN
RELATED QUESTIONS
Find the zeros of the quadratic polynomial 6x2 - 13x + 6 and verify the relation between the zero and its coefficients.
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 a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, − 7, − 14 respectively
Find all zeroes of the polynomial `(2x^4 - 9x^3 + 5x^2 + 3x - 1)` if two of its zeroes are `(2 + sqrt3)` and `(2 - sqrt3)`
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
`p(x) = x^2 + 2sqrt2x + 6`
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 (x+a) is a factor of the polynomial `2x^2 + 2ax + 5x + 10`, find the value of a.
Find the quotient and the remainder when f(x) = x4 – 5x + 6 is divided by g(x) = 2 – x2.
On dividing 3x3 + x2 + 2x + 5 is divided by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).
If α, β are the zeros of the polynomial f(x) = ax2 + bx + c, then\[\frac{1}{\alpha^2} + \frac{1}{\beta^2} =\]
If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are each equal to zero, then the third zero is
If two zeros x3 + x2 − 5x − 5 are \[\sqrt{5}\ \text{and} - \sqrt{5}\], then its third zero is
A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is
If x + 2 is a factor of x2 + ax + 2b and a + b = 4, then
The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is
A quadratic polynomial, whose zeroes are –3 and 4, is ______.
If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
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.
`(-8)/3, 4/3`
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.
