Advertisements
Advertisements
Question
If the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.
Advertisements
Solution
Let α = a - d, β = a and γ = a + d be the zeros of the polynomial
f(x) = 2x3 − 15x2 + 37x − 30
Therefore
`alpha+beta+gamma=("coefficient of "x^2)/("coefficient of "x^3)`
`=-((-15)/2)`
`=15/2`
`alphabetagamma="-constant term"/("coefficient of "x^2)`
`=-((-30)/2)`
= 15
Sum of the zeroes `=("coefficient of "x^2)/("coefficient of "x^3)`
`(a-d)+a+(a+d)=15/2`
`a+a+a-d+d=15/2`
`3a=15/2`
`a=15/2xx1/3`
`a=5/2`
Product of the zeroes `="-constant term"/("coefficient of "x^2)`
`alphabetagamma=15`
`(a-d)+a+(a+d)=15`
`a(a^2-d^2)=15`
Substituting a = 5/2 we get
`5/2((5/2)^2-d^2)=15`
`5/2(25/4-d^2)=15`
`25/4-d^2=15xx2/5`
`25/4-d^2=3xx2`
`25/4-d^2=6`
`-d^2=6-25/4`
`-d^2=(24-25)/4`
`-d^2=(-1)/4`
`d^2=1/4`
`d xx d=1/2xx1/2`
`d=1/2`
Therefore, substituting a=5/2 and d=1/2 in α = a - d, β = a and γ = a + d
α = a - d
`alpha=5/2-1/2`
`alpha=(5-1)/2`
`alpha=4/2`
`alpha=2`
β = a
`beta=5/2`
γ = a + d
`gamma=5/2+1/2`
`gamma=(5+1)/2`
`gamma=6/2`
`gamma=3`
Hence, the zeros of the polynomial are 2, `5/2` , 3.
APPEARS IN
RELATED QUESTIONS
Find the zeros of the quadratic polynomial 6x2 - 13x + 6 and verify the relation between the zero and its coefficients.
Find the zeros of the quadratic polynomial 4x2 - 9 and verify the relation between the zeros and its coffiecents.
Find the zeros of the quadratic polynomial 9x2 - 5 and verify the relation between the zeros and its coefficients.
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`0, sqrt5`
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 α2β + αβ2
If α and β are the zeros of the quadratic polynomial f(x) = 6x2 + x − 2, find the value of `alpha/beta+beta/alpha`.
If α and β are the zeroes of the polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α − β)2.
Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
Find a cubic polynomial whose zeroes are 2, -3and 4.
If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]
If α, β are the zeros of the polynomial f(x) = ax2 + bx + c, then\[\frac{1}{\alpha^2} + \frac{1}{\beta^2} =\]
What should be subtracted to the polynomial x2 − 16x + 30, so that 15 is the zero of the resulting polynomial?
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
Case Study -1
The figure given alongside shows the path of a diver, when she takes a jump from the diving board. Clearly it is a parabola.
Annie was standing on a diving board, 48 feet above the water level. She took a dive into the pool. Her height (in feet) above the water level at any time ‘t’ in seconds is given by the polynomial h(t) such that h(t) = -16t2 + 8t + k.
The zeroes of the polynomial r(t) = -12t2 + (k - 3)t + 48 are negative of each other. Then k is ______.
Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1?
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:
`2x^2 + (7/2)x + 3/4`
Find the sum and product of the roots of the quadratic equation 2x2 – 9x + 4 = 0.
