Question

If 2 and –2 are two zeros of the polynomial 2x⁴ – 5x³ – 11x² + 20x + 12, find all the zeros of the given polynomial.

Answer 1

Given: 2x⁴ – 5x³ – 11x² + 20x + 12.

Step-wise calculation:

1. Since 2 and –2 are zeros, (x – 2)(x + 2) = x² – 4 is a factor. 

Write 2x⁴ – 5x³ – 11x² + 20x + 12 = (x² – 4) × Q(x), where Q(x) is a quadratic ax² + bx + c.

2. Multiply out (x² – 4)(ax² + bx + c) = ax⁴ + bx³ + (c – 4a)x² – 4bx – 4c. 

Equate coefficients with 2x⁴ – 5x³ – 11x² + 20x + 12:

a = 2

b = –5

c – 4a = –11

⇒ c – 8 = –11

⇒ c = –3

Check: –4b = 20 ⇒ b = –5 (consistent); –4c = 12 ⇒ c = –3 (consistent). 

Thus, Q(x) = 2x² – 5x – 3.

3. Factor Q(x): 2x² – 5x – 3 = (2x + 1)(x – 3).

So, full factorization: 2x⁴ – 5x³ – 11x² + 20x + 12 = (x – 2)(x + 2)(2x + 1)(x – 3).

The zeros are x = `2, -2, -1/2, 3`.