Advertisements
Advertisements
प्रश्न
If α and β are the roots of the polynomial f(x) = x2 – 2x + 3, find the polynomial whose roots are `(alpha - 1)/(alpha + 1), (beta - 1)/(beta + 1)`
Advertisements
उत्तर
Sum of the roots
= `(alpha - 1)/(alpha + 1) + (beta - 1)/(beta + 1)`
= `((alpha - 1)(beta + 1) + (beta - 1)(alpha + 1))/((alpha + 1)(beta + 1))`
= `(alphabeta + alpha - beta - 1 + alphabeta + beta - alpha - 1)/(alphabeta + alpha + beta + 1)`
= `(2alphabeta - 2)/(alphabeta + alpha+ beta + 1)`
= `(2(3) - 2)/(3 + 2 + 1)`
= `4/6`
= `2/3`
Product of the roots
= `(alpha - 1)/(alpha + 1) xx (beta - 1)/(beta + 1)`
= `((alpha - 1)(beta - 1))/((alpha + 1)(beta + 1))`
= `(alphabeta - alpha - beta + 1)/(alphabeta + alpha + beta + 1)`
= `(alphabeta - (alpha + beta) + 1)/(alphabeta + (alpha + beta) + 1)`
= `(3 - 2 + 1)/(3 + 2 + 1)`
= `2/6`
= `1/3`
The quadratic polynomial is
x2 – (sum of the roots) x + products of the roots = 0
`x^2 – (2/3) x + 1/3` = 0
3x2 – 2x + 1 = 0
APPEARS IN
संबंधित प्रश्न
Find the square root of the following rational expression
`(400x^4y^12z^16)/(100x^8y^4z^4)`
Find the square root of the following
4x2 + 20x + 25
Find the square root of the following
9x2 – 24xy + 30xz – 40yz + 25z2 + 16y2
Find the square root of the following
`(2x^2 + 17/6 x + 1) (3/2 x^2 + 4x + 2) (4/3 x^2 + 11/3 x + 2)`
Find the square root of the following polynomials by division method
x4 – 12x3 + 42x2 – 36x + 9
Find the square root of the following polynomials by division method
37x2 – 28x3 + 4x4 + 42x + 9
If the roots of the equation q2x2 + p2x + r2 = 0 are the squares of the roots of the equation qx2 + px + r = 0, then q, p, r are in __________
Find the square root of 289x4 – 612x3 + 970x2 – 684x + 361
Solve `sqrt(y + 1) + sqrt(2y - 5)` = 3
If α and β are the roots of the polynomial f(x) = x2 – 2x + 3, find the polynomial whose roots are α + 2, β + 2
