Question

If 1/2 and 1 are zeros of 2x^3+x^2-5x+2 find the other zeros

Answer 1

Hey friend!

Here is your answer:

To find the other zeros of 2x³ + x² - 5x  + 2 when ¹/₂ and 1 are the two zeroes of the polynomial

Information:

One degree polynomial ↔ one zero

Two degree polynomial ↔ two zeros

Three degree polynomial ↔ three zeros

Hence, the polynomial has three zeros because it's degree is 3

Let p(x) =  2x³ + x² - 5x  + 2

If ¹/₂ is a zero of the polynomial p(x)

x = ¹/₂ ⇒ x - ¹/₂

then,

⇒ x - ¹/₂ is a factor of p(x)

Similarly as 1 is a zero of p(x)

⇒ x - 1 is a factor of p(x)

Now,

(x  - ¹/₂)(x - 1) is a factor of p(x)

⇒ x² - ( ¹/₂ + 1)x +  ¹/₂ is a factor of p(x)

⇒ x² - ³/₂x +  ¹/₂  is a factor of p(x)

On dividing p(x) = 2x³ + x² - 5x  + 2 by the factor x² - ³/₂x +  ¹/₂,

we must get another factor of p(x) [ of one degree]

[Calculation]

Quotient/Factor = 2x + 4

Hence,

2x + 4 = 0

⇒ 2x = - 4

⇒ x = - 2

∴ The other zero of polynomial  2x³ + x² - 5x  + 2 is - 2

I hope it helps you#QualityAnswer