Verify that $3, - 2, 1$ are the zeroes of the cubic polynomial $p (x) = (x^{3} - 2 x^{2} - 5 x + 6)$ and verify the relationship between its zeroes and coefficients.

Open in App

Solution

Let $f (x) = x^{3} - 2 x^{2} - 5 x + 6$
$3, - 2$ and $1$ are the zeroes of the polynomial ( given )
Therefore,
$f (3) = (3)^{3} - 2 (3)^{2} - 5 (- 2) + 6$
$= 27 - 18 - 15 + 6$
$= 0$
$f (- 2) = (- 2)^{3} - 2 (- 2)^{2} - 5 (- 2) + 6$
$- 8 - 8 + 10 + 6$
$= 0$
$f (1) = (1)^{3} - 2 (1)^{2} - 5 (1) + 6$
$= 1 - 2 - 5 + 6$
$= 0$
Verify relations :
General form of cubic equation : $a x^{3} + b x^{2} + c x + d$
now ,
Consider $α = 3, β = - 2$ and $y = 1$
$α + β + y = 3 - 2 + 1 = 2 = - b / a$
$α β + β y + α y = 3 (- 2) + (- 2) (1) + 1 (3) = - 5 = c / a$
and $α β y = 3 (- 2) (1) = - 6 = - d / a$

Suggest Corrections

Similar questions

Verify that 3, -2, 1 are the zeros of the cubic polynomial $p (x) = x^{3} - 2 x^{2} - 5 x + 6$ and verify the relation betweeen its zeros and coefficients.

Q. Verity that 3, −2, 1 are the zeros of the cubic polynomial p(x) = x³ − 2x² − 5x + 6 and verify the relation between its zeros and coefficients.

Q. 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:
(i)

2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2

(ii)

x^{3} - 4 x^{2} + 5 x - 2; 2, 1, 1

Verify that the numbers given along side of the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and coefficients in in each case:

(i) $f (x) = 2 x^{3} + x^{2} - 5 x + 2; \frac{1}{2}, 1, - 2$

(ii) $g (x) = x^{3} - 4 x^{2} + 5 x - 2; 2, 1, 1$

Q. Verify that 3, -2, 1 are the zeroes of the cubic polynomial

p (x) = x^{3} - 2 x^{2} - 5 x + 6

Join BYJU'S Learning Program

Verify that 3,−2,1 are the zeroes of the cubic polynomial p(x)=(x3−2x2−5x+6) and verify the relationship between its zeroes and coefficients.

Verify that $3, - 2, 1$ are the zeroes of the cubic polynomial $p (x) = (x^{3} - 2 x^{2} - 5 x + 6)$ and verify the relationship between its zeroes and coefficients.