Question

If -1 and 2 are the zeroes of the polynomial 2x³-x²-5x-2, find it's third zero

Answer 1

Answer:

- 1/2

Step-by-step explanation:

p(x) = 2x³ - x² - 5x - 2

-1 and 2 are two zeroes of p(x).

∴ (x + 1), (x - 2) are two factors of p(x).

Let the third zero be a.

In p(x), the coefficient of x³ is 2. In the factors (x + 1) and (x - 2), the coefficient of x is 1 each.

The coefficient of x in the third factor is the coefficient of x³ in p(x) divided by the product of coefficients of x in the other two factors.

So the coefficient of x in third factor will be,

2 ÷ (1 × 1) = 2

So it is 2x in the third factor.

As a is the third zero, '2a' should be subtracted from 2x to get a as zero.

∴ Let the third factor be (2x - 2a).

Okay, let me write the following:

p(x) = 2x³ - x² - 5x - 2                            →    (1)

p(x) = (x + 1)(x - 2)(2x - 2a)

p(x) = (x² - x - 2)(2x - 2a)

p(x) = 2x³ - 2ax² - 2x² +2ax - 4x + 4a

p(x) = 2x³ - 2(a + 1) x² + 2(a - 2) x + 4a    →    (2)

Here, (1) = (2)

From (1) and (2), take the coefficients of x^0 each, i.e., -2 and 4a.

4a = - 2

a = - 2 / 4

a = - 1 / 2

∴ - 1/2 is the third root.

And (2x + 1) is the third factor.

Thank you. Have a nice day. :-)

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