Advertisements
Advertisements
рдкреНрд░рд╢реНрди
If If α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3, find a polynomial whose roots are α + 2, β + 2.
Advertisements
рдЙрддреНрддрд░
Since α and β are the zeros of the quadratic polynomial f(x) = x2 – 2x + 3
`alpha+beta="-coefficient of x"/("coefficient of "x^2)`
`=(-(-2))/1`
= 2
Product of the zeroes `="constant term"/("coefficient of "x^2)`
`=3/1`
= 3
Let S and P denote respectively the sums and product of the polynomial whose zeros
α + 2, β + 2
S = (α + 2) + (β + 2)
S = α + β + 2 + 2
S = 2 + 2 + 2
S = 6
P = (α + 2)(β + 2)
P = αβ + 2β + 2α + 4
P = αβ + 2(α + β) + 4
P = 3 + 2(2) + 4
P = 3 + 4 + 4
P = 11
Therefore the required polynomial f(x) is given by
f(x) = k(x2 - Sx + P)
f(x) = k(x2 - 6x + 11)
Hence, the required equation is f(x) = k(x2 - 6x + 11)
рд╕рдВрдмрдВрдзрд┐рдд рдкреНрд░рд╢реНтАНрди
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
`1/4 , -1`
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
1, 1
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case
x3 – 4x2 + 5x – 2; 2, 1, 1
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α - β
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `beta/(aalpha+b)+alpha/(abeta+b)`
If ЁЭЫ╝ and ЁЭЫ╜ are the zeros of the quadratic polynomial f(x) = x2 − 5x + 4, find the value of `1/alpha+1/beta-2alphabeta`
Find the zeroes of the quadratic polynomial `4x^2 - 4x + 1` 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.
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 a cubic polynomial whose zeroes are `1/2, 1 and -3.`
On dividing `3x^3 + x^2 + 2x + 5` is divided by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).
What should be subtracted to the polynomial x2 − 16x + 30, so that 15 is the zero of the resulting polynomial?
If two zeroes of the polynomial x3 + x2 − 9x − 9 are 3 and −3, then its third zero is
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:
5t2 + 12t + 7
The zeroes of the quadratic polynomial x2 + 99x + 127 are ______.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`v^2 + 4sqrt(3)v - 15`
If α, β are zeroes of the quadratic polynomial x2 – 5x + 6, form another quadratic polynomial whose zeroes are `1/α, 1/β`.
The zeroes of the polynomial p(x) = 25x2 – 49 are ______.
If α, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of α2 + β2.
