Advertisements
Advertisements
प्रश्न
Find all zeros of the polynomial 2x4 + 7x3 − 19x2 − 14x + 30, if two of its zeros are `sqrt2` and `-sqrt2`.
उत्तर
We know that if x = a is a zero of a polynomial, then x - a is a factor of f(x).
Since, `sqrt2` and `-sqrt2` are zeros of f(x).
Therefore
`(x+sqrt2)(x-sqrt2)=x^2-(sqrt2)^2`
= x2 - 2
x2 - 2 is a factor of f(x). Now, we divide 2x4 + 7x3 − 19x2 − 14x + 30 by g(x) = x2 - 2 to find the zero of f(x).
By using division algorithm we have
f(x) = g(x) x q(x) - r(x)
2x4 + 7x3 − 19x2 − 14x + 30 = (x2 - 2)(2x2 + 7x - 15) + 0
2x4 + 7x3 − 19x2 − 14x + 30 `=(x+sqrt2)(x-sqrt2)(2x^2+10x-3x-15)`
2x4 + 7x3 − 19x2 − 14x + 30 `=(x+sqrt2)(x-sqrt2)[2x(x+5)-3(x+5)]`
2x4 + 7x3 − 19x2 − 14x + 30 `=(x+sqrt2)(x-sqrt2)(2x-3)(x+5)`
Hence, the zeros of the given polynomial are `-sqrt2`, `+sqrt2`, `(+3)/2`, -5.
APPEARS IN
संबंधित प्रश्न
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial
x2 + 3x + 1, 3x4 + 5x3 – 7x2 + 2x + 2
On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x)
Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm
deg q(x) = deg r(x)
If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a, find k and a.
Find all zeros of the polynomial f(x) = 2x4 − 2x3 − 7x2 + 3x + 6, if its two zeros are `-sqrt(3/2)` and `sqrt(3/2)`
What must be added to the polynomial f(x) = x4 + 2x3 − 2x2 + x − 1 so that the resulting polynomial is exactly divisible by x2 + 2x − 3 ?
Find all the zeros of the polynomial 2x3 + x2 − 6x − 3, if two of its zeros are `-sqrt3` and `sqrt3`
If `x^3+ x^2-ax + b` is divisible by `(x^2-x)`,write the value of a and b.
Find the quotient and remainder of the following.
(8y3 – 16y2 + 16y – 15) ÷ (2y – 1)
The area of a rectangle is x2 + 7x + 12. If its breadth is (x + 3), then find its length
