Advertisements
Advertisements
प्रश्न
Given that `sqrt(2)` is a zero of the cubic polynomial `6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2)`, find its other two zeroes.
Advertisements
उत्तर
Given, `sqrt(2)` is one of the zero of the cubic polynomial.
Then, `(x - sqrt(2))` is one of the factor of the given polynomial p(x) = `6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2)`.
So, by dividing p(x) by `x - sqrt(2)`
`6x^2 + 7sqrt(2)x + 4`
`(x - sqrt(2))")"overline(6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2))`
`6x^3 - 6sqrt(2)x^2`
– +
`7sqrt(2)x^2 - 10x - 4sqrt(2)`
`7sqrt(2)x^2 - 14x`
– +
`4x - 4sqrt(2)`
`4x - 4sqrt(2)`
0
`6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2) = (x - sqrt(2)) (6x^2 + 7sqrt(2)x + 4)`
By splitting the middle term,
We get,
`(x - sqrt(2)) (6x^2 + 4sqrt(2)x + 3sqrt(2)x + 4)`
= `(x - sqrt(2)) [2x(3x + 2sqrt(2)) + sqrt(2)(3x + 2sqrt(2))]`
= `(x - sqrt(2)) (2x + sqrt(2)) (3x + 2sqrt(2))`
To get the zeroes of p(x),
Substitute p(x) = 0
`(x - sqrt(2)) (2x + sqrt(2)) (3x + 2sqrt(2))` = 0
`x = sqrt(2) , x = -sqrt(2)/2, x = (-2sqrt(2))/3`
Hence, the other two zeroes of p(x) are `-sqrt(2)/2` and `(-2sqrt(2))/3`.
APPEARS IN
संबंधित प्रश्न
Prove relation between the zeros and the coefficient of the quadratic polynomial ax2 + bx + c
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
6x2 – 3 – 7x
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
4u2 + 8u
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`0, sqrt5`
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
`-1/4 ,1/4`
If the squared difference of the zeros of the quadratic polynomial f(x) = x2 + px + 45 is equal to 144, find the value of p.
If the zeros of the polynomial f(x) = x3 − 12x2 + 39x + k are in A.P., find the value of k.
Find the zeroes of the polynomial f(x) = `2sqrt3x^2-5x+sqrt3` and verify the relation between its zeroes and coefficients.
Find the quadratic polynomial whose zeroes are `2/3` and `-1/4`. Verify the relation between the coefficients and the zeroes of the polynomial.
Find the quadratic polynomial, sum of whose zeroes is `sqrt2` and their product is `(1/3)`.
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 two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are each equal to zero, then the third zero is
If \[\sqrt{5}\ \text{and} - \sqrt{5}\] are two zeroes of the polynomial x3 + 3x2 − 5x − 15, then its third zero is
Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x),
where p(x) = x5 − 4x3 + x2 + 3x +1, g(x) = x3 − 3x + 1
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 ______.
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:
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
4x2 – 3x – 1
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 one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is reciprocal of the other, then find the value of k.
