Question

Using factor theorem, factorize each of the following polynomial:
2y³+y²-2y-1

Answer 1

GIVEN :

Polynomial = 2y³ + y² - 2y - 1.

TO FIND :

The factors of the polynomial using factor theorem.

SOLUTION :

• Let the given polynomial be denoted as f(y) for calculations.

• Factor theorem states that if f(a) = 0, then (a - 1) is a factor of the polynomial f(y). This method is based on "hit and trial" process.

• Let us start with 1.

Putting y = 1, we get,

f(1) = 2.1³ + 1² - 2.1 - 1

Or, f(1) = 2.1 + 1 - 2 - 1

Or, f(1) = 2 + 1 - 2 - 1

Or, f(1) = 0

• Therefore, y = 1.

Or, y - 1 = 0

• Since f(1) = 0, (y - 1) is a factor of f(y). So, the other factors of f(y) can be obtained by dividing f(y) by (y - 1).

• The quotient obtained is 2y² + 3y + 1 ( refer to the image attached below for division ).

• Now, 2y² + 3y + 1 is in a quadratic form, which can be factorized further;

2y² + 3y + 1

= 2y² + 2y + y + 1

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

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

• Therefore, the factors of the polynomial f(y) can be written as (y - 1) (y + 1) (2y + 1).

Answer - The polynomial 2y³ + y² - 2y - 1 can be factorized into (y - 1) (y + 1) (2y + 1) using factor theorem.