Using factor theorem, factorize each of the following polynomials:$2y^3+ y^2 - 2y - 1$

Given:

Given expression is $2y^3+ y^2 - 2y - 1$.

To do:

We have to find the given polynomial using factor theorem.

Solution:

Let $f(y) = 2y^3 + y^2 - 2y - 1$

The factors of the constant term $-1$ are $\pm 1$.

The factor of the coefficient of $y^3$ is $2$. 

Therefore, the possible rational roots are $\pm 1, \pm \frac{1}{2}$

$f (1) = 2 (1)^3 + (1)^2 - 2 (1) - 1$

$= 2 + 1 - 2 - 1$

$= 0$

Therefore, $(y - 1)$ is a factor of $f(y)$

Dividing $f(y)= 2y3 + y2 - 2y - 1$ by $(y - 1)$, we get,

$2y^3 + y^2 - 2y - 1 = (y - 1) (2y^2 + 3y + 1)$

$2y2 + 3y + 1= 2y^2 + 2y + y + 1$

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

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

Hence, $2y^3 + y^2 - 2y - 1 = (y - 1) (2y + 1) (y + 1)$.

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Updated on: 10-Oct-2022

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