Question

Using factor theorem, factorize each of the following polynomial:

2y³ + y² - 2y - 1

Answer 1

Let, f (y) = 2y³ + y² - 2y - 1

The factors of the constant term - 1 are \(\pm\) 1

The factor of the coefficient of y³ is 2. 

Hence, 

possible rational roots are \(\pm1\pm\frac{1}{2}\)

We have,

f (1) = 2 (1)³ + (1)² – 2 (1) - 1

= 2 + 1 – 2 - 1

= 0

So, 

(y - 1) is a factor of f (y)

Let us now divide

f (y) = 2y³ + y² - 2y - 1 by (y - 1) to get the other factors of f (x)

Using long division method, we get

2y³ + y² - 2y - 1 = (y - 1) (2y² + 3y + 1)

2y² + 3y + 1 = 2y² + 2y + y + 1

= 2y (y + 1) + 1 (y + 1)

= (2y + 1) (y + 1)

Hence, 

2y³ + y² - 2y - 1 = (y - 1) (2y + 1) (y + 1)