Advertisements
Advertisements
Question
If the zeros of the polynomial f(x) = x3 − 12x2 + 39x + k are in A.P., find the value of k.
Advertisements
Solution
Let a - d, a and a + d be the zeros of the polynomial f(x). Then,
Sum of the zeroes `=("coefficient of "x^2)/("coefficient of "x^3)`
`a-d+a+a+d=(-(-12))/1`
`a-d+a+a+d=12`
`3a=12`
`a=12/3=4`
Since 'a' is a zero of the polynomial f(x).
f(x) = x3 − 12x2 + 39x + k
f(a) = 0
f(a) = 43 − 12(4)2 + 39(4) + k
0 = 64 - 192 + 156 + k
0 = 220 - 192 + k
0 = 28 + k
-28 = k
Hence, the value of k is -28.
APPEARS IN
RELATED QUESTIONS
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 the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
x2 – 2x – 8
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
4, 1
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
`f(x)=x^2-(sqrt3+1)x+sqrt3`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate :
`a(α^2/β+β^2/α)+b(α/β+β/α)`
If 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x2 − 5x −1, find the value of α2β + αβ2.
If α and β are the zeros of the quadratic polynomial f(x) = x2 − p (x + 1) — c, show that (α + 1)(β +1) = 1− c.
Find the quadratic polynomial, sum of whose zeroes is `( 5/2 )` and their product is 1. Hence, find the zeroes of the polynomial.
If 1 and –2 are two zeroes of the polynomial `(x^3 – 4x^2 – 7x + 10)`, find its third zero.
The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is
Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial ______.
Basketball and soccer are played with a spherical ball. Even though an athlete dribbles the ball in both sports, a basketball player uses his hands and a soccer player uses his feet. Usually, soccer is played outdoors on a large field and basketball is played indoor on a court made out of wood. The projectile (path traced) of soccer ball and basketball are in the form of parabola representing quadratic polynomial.
What will be the expression of the polynomial?
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:
`4x^2 + 5sqrt(2)x - 3`
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.
`(-3)/(2sqrt(5)), -1/2`
Find the zeroes of the quadratic polynomial 6x2 – 3 – 7x and verify the relationship between the zeroes and the coefficients.
If p(x) = x2 + 5x + 6, then p(– 2) is ______.
If α, β are zeroes of the quadratic polynomial x2 – 5x + 6, form another quadratic polynomial whose zeroes are `1/α, 1/β`.
If α, β are zeroes of quadratic polynomial 5x2 + 5x + 1, find the value of α–1 + β–1.
