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 rational expression
`(121("a" + "b")^8 (x + y)^8 ("b" - "c")^8)/(81("b" - "c")^4 ("a" - "b")^12 ("b" - "c")^4)`
Find the square root of the following
(4x2 – 9x + 2)(7x2 – 13x – 2)(28x2 – 3x – 1)
Find the square root of the following polynomials by division method
37x2 – 28x3 + 4x4 + 42x + 9
Find the square root of the following polynomials by division method
16x4 + 8x2 + 1
Find the square root of the following polynomials by division method
121x4 – 198x3 – 183x2 + 216x + 144
Find the values of a and b if the following polynomial is a perfect square
ax4 + bx3 + 361x2 + 220x + 100
Find the values of m and n if the following polynomial is a perfect square
`1/(x^4) - 6/(x^3) + 13/(x^2) + "m"/x + "n"`
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 __________
If – 4 is a root of the equation x2 + px – 4 = 0 and if the equation x2 + px + q = 0 has equal roots, find the values of p and q.
