Verified answer

2x^4 + x³ - 9x² - 4x + 4

You try to find an easy root: - 2; - 1; 1 ; 2

You can see that - 1 is a root, because the polynomial is null → you factorize (x + 1)

= (x + 1)(2x³ + ax² + bx + c) → you expand

= 2x^4 + ax³ + bx² + cx + 2x³ + ax² + bx + c → you group

= 2x^4 + x³(a + 2) + x²(b + a) + x(c + b) + c → you compare to: 2x^4 + x³ - 9x² - 4x + 4

c = 4

(c + b) = - 4 → b = - 4 - c → b = - 8

(b + a) = - 9 → a = - 9 - b → a = - 1

(a + 2) = 1 → a = - 1 (of course)

= (x + 1)(2x³ - x² - 8x + 4)

Let's look the polynomial: (2x³ - x² - 8x + 4)

You try to find an easy root: - 2; - 1; 1 ; 2

You can see that - 2 is a root, because the polynomial is null → you factorize (x + 2)

= (x + 2)(2x² + ax + b) → you expand

= 2x³ + ax² + bx + 4x² + 2ax + 2b → you group

= 2x³ + x²(a + 4) + x(b + 2a) + 2b → you compare to: 2x³ - x² - 8x + 4

2b = 4 → b = 2

(b + 2a) = - 8 → 2a = - 8 - b → 2a = - 10 → a = - 5

(a + 4) = - 1 → a = - 5 (of course)

= (x + 2)(2x² - 5x + 2)

Let's look the polynomial: (2x² - 5x + 2)

Polynomial like: ax² + bx + c, where:

a = 2

b = - 5

c = 2

Δ = b² - 4ac (discriminant)

Δ = (- 5)² - 4(2 * 2) = 25 - 16 = 9 = 3²

x1 = (- b - √Δ) / 2a = (5 - 3) / (2 * 2) = 2/4 = 1/2

x2 = (- b + √Δ) / 2a = (5 + 3) / (2 * 2) = 8/4 = 2

The polynomial can be written like: a(x - x1)(x - x2)

2x² - 5x + 2 = 2 * [x - (1/2)] * (x - 2)

2x² - 5x + 2 = (2x - 1)(x - 2)

Resume:

= 2x^4 + x³ - 9x² - 4x + 4

= (x + 1)(2x³ - x² - 8x + 4)

= (x + 1)(x + 2)(2x² - 5x + 2)

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

The roots are: - 1, - 2, 1/2, 2

2x^4+x³-9x²-4x+4 =

2x²(x² -4) + x(x²- 4) - (x² - 4) =

(x² -4)(2x² + x - 1) =

(x² -4)(2x-1)(x+1)

Roots are -2, 2, 1/2, -1