Advertisements
Advertisements
प्रश्न
If 3 and –3 are two zeroes of the polynomial `(x^4 + x^3 – 11x^2 – 9x + 18)`, find all the zeroes of the given polynomial.
Advertisements
उत्तर
Let `x^4 + x^3 – 11x^2 – 9x + 18`
Since 3 and – 3 are the zeroes of f(x), it follows that each one of (x + 3) and (x – 3) is a factor of f(x).
Consequently, `(x – 3) (x + 3) = (x^2 – 9)` is a factor of f(x).
On dividing `f(x) by (x^2 – 9)`, we get:
`f(x) = 0 ⇒ (x^2 + x – 2) (x^2 – 9) = 0`
⇒ `(x^2 + 2x – x – 2) (x – 3) (x + 3)`
⇒ `(x – 1) (x + 2) (x – 3) (x + 3) = 0`
⇒ `x = 1 or x = -2 or x = 3 or x = -3`
Hence, all the zeroes are 1, -2, 3 and -3.
APPEARS IN
संबंधित प्रश्न
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
`p(x) = x^2 + 2sqrt2x + 6`
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) = ax2 + bx + c, then evaluate :
`a(α^2/β+β^2/α)+b(α/β+β/α)`
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 quadratic polynomial `2x^2 ˗ 11x + 15` and verify the relation between the zeroes and the 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.
If `x =2/3` and x = -3 are the roots of the quadratic equation `ax^2+2ax+5x ` then find the value of a and b.
Verify that 5, -2 and 13 are the zeroes of the cubic polynomial `p(x) = (3x^3 – 10x^2 – 27x + 10)` and verify the relation between its zeroes and coefficients.
Find a cubic polynomial whose zeroes are `1/2, 1 and -3.`
If f(x) =` x^4 – 3x^2 + 4x + 5` is divided by g(x)= `x^2 – x + 1`
If 1 and –2 are two zeroes of the polynomial `(x^3 – 4x^2 – 7x + 10)`, find its third zero.
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 added to the polynomial x2 − 5x + 4, so that 3 is the zero of the resulting polynomial?
If two zeroes of the polynomial x3 + x2 − 9x − 9 are 3 and −3, then its third zero is
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 ______.
If one of the zeroes of the cubic polynomial x3 + ax2 + bx + c is –1, then the product of the other two zeroes is ______.
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.
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.
If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is reciprocal of the other, then find the value of k.
