Question

If -1,2 are the two zeroes of the polynomial 2xcube -xsquare -5x-2,find the third zero​

Answer 1

Solution :-

Let f(x) = 2x³ - x² - 5x - 2

- 1, 2 are the 2 zeroes of the polynomial

Therefore (x + 1) , (x - 2) are the factors of f(x).

(x + 1)(x - 2) = x(x - 2) + 1(x - 2) = x² - 2x + x - 2 = x² - x - 2

x² - x - 2 is the factor of f(x)

So, divide f(x) by x² - x - 2

[ Refer to attachment ]

Hence, f(x) = 2x³ - x² - 5x - 2 = (2x + 1)(x² - x - 2)

2x + 1 is a factor of f(x)

Equate 2x + 1 to 0 to find zero

⇒ 2x + 1 = 0

⇒ 2x = - 1

⇒ x = - 1/2

Therefore, the 3rd zero is - 1/2.

Answer 2

Given :-----

(-1,2) are the two zeroes of the polynomial 2x³ -x² -5x -2 .

To Find :----

Third zero of polynomial ?

Concept used :------

we can say that x = a is a root or zero of a polynomial if it is a solution to the equation P(x) = 0 ....

Solution :------

Given (-1 and 2 ) are the zeros of polynomial ,

Hence,

p(-1) = 0

→ X = -1

→ x+1 = 0

similarly

→ (x -2) = 0

so, (x+1) and (x-2) are the factors of Given polynomial .

or, we can say that,

→ (x+1)(x-2) = x² -2x+x-2 = (x² -x -2) is a factor of polynomial ..

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Now, lets divide the polynomial to Find third zero..

x² -x -2 ) 2x³ -x² -5x -2 ( 2x +1

- 2x³-2x²-4x

x²-x-2

- x²-x-2

__0___

Hence, our third Root will be ,,,,

→ 2x+1 = 0

→ 2x = (-1)

→ x = (-1/2) (Ans) ...

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Extra brainly knowledge :-----

The general form of a cubic equation is ax³ + bx² + cx + d = 0 where a, b, c and d are constants and a ≠ 0.

❁ sum of roots = -b/a

❁ product of roots = -d/a

❁ the sum of the product of roots taken two at a time = c/a

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