Question
Using factor theorem, factorize the following polynomials: $2y^3 - 5y^2 - 19y + 42$
✓
Answer
Let $f(y)=2 y^3-5 y^2-19 y+42$ be the given polynomial.
Now, putting $y =2$,
we get $f(2) = 2(2)^3 - 5(2)^2 - 19(2) + 42 $
$= 16 - 20 - 38 + 42 = -58 + 58 = 0$
Therefore, $(y-2)$ is a factor of polynomial $f(y)$.
Now, $f(y) = 2y^2(y - 2) - y(y - 2) - 21(y - 2) $
$= (y - 2)(2y^2 - y - 21) $
$= (y - 2)(2y^2 - 7y + 6y - 21)$
$ = (y - 2)(y + 3)(2y - 7)$
Hence $(y-2),(y+3)$ and $(2 y-7)$ are the factors of polynomial $f(y)$.
Now, putting $y =2$,
we get $f(2) = 2(2)^3 - 5(2)^2 - 19(2) + 42 $
$= 16 - 20 - 38 + 42 = -58 + 58 = 0$
Therefore, $(y-2)$ is a factor of polynomial $f(y)$.
Now, $f(y) = 2y^2(y - 2) - y(y - 2) - 21(y - 2) $
$= (y - 2)(2y^2 - y - 21) $
$= (y - 2)(2y^2 - 7y + 6y - 21)$
$ = (y - 2)(y + 3)(2y - 7)$
Hence $(y-2),(y+3)$ and $(2 y-7)$ are the factors of polynomial $f(y)$.
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