Advertisements
Advertisements
Question
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.
`-2sqrt(3), -9`
Advertisements
Solution
Sum of the zeroes = `- 2sqrt(3)`
Product of the zeroes = – 9
P(x) = x2 – (Sum of the zeroes) + (Product of the zeroes)
Then, P(x) = `x^2 - (-2sqrt(3)x) - 9`
Using splitting the middle term method,
`x^2 + 2sqrt(3)x - 9` = 0
`x^2 + (3sqrt(3)x - sqrt(3)x) - 9` = 0
`x(x + 3sqrt(3)) - sqrt(3)(x + 3sqrt(3))` = 0
`(x - sqrt(3))(x + 3sqrt(3))` = 0
`\implies` x = `sqrt(3), -3sqrt(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 zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
t2 – 15
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 the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.
Find the quadratic polynomial, sum of whose zeroes is 0 and their product is -1. Hence, find the zeroes of the polynomial.
Find a cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time and the product of its zeroes as 5, -2 and -24 respectively.
If f(x) = `x^4– 5x + 6" is divided by g(x) "= 2 – x2`
If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]
What should be subtracted to the polynomial x2 − 16x + 30, so that 15 is the zero of the resulting polynomial?
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 ______.
An asana is a body posture, originally and still a general term for a sitting meditation pose, and later extended in hatha yoga and modern yoga as exercise, to any type of pose or position, adding reclining, standing, inverted, twisting, and balancing poses. In the figure, one can observe that poses can be related to representation of quadratic polynomial.
The zeroes of the quadratic polynomial `4sqrt3"x"^2 + 5"x" - 2sqrt3` are:
If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the product of the other two zeroes is ______.
If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
If all three zeroes of a cubic polynomial x3 + ax2 – bx + c are positive, then at least one of a, b and c is non-negative.
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`
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.
`(-8)/3, 4/3`
Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.
If the zeroes of the polynomial x2 + px + q are double in value to the zeroes of the polynomial 2x2 – 5x – 3, then find the values of p and q.
If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is reciprocal of the other, then find the value of k.
Find a quadratic polynomial whose zeroes are 6 and – 3.
