Using factor theorem, factorize each of the following polynomial:
y3-7y+ 6
Let, f (y) = y3-7y+ 6
The constant term in f (y) is equal to + 6 and factors of + 6 are ,
Putting y = 1 in f (y), we have
f (1) = (1)3 – 7 (1) + 6
= 1 – 7 + 6
= 0
Therefore, (y - 1) is a factor of f (y).
Similarly, (y - 2) and (y + 3) are the factors of f (y).
Since, f (y) is a polynomial of degree 3. So, it cannot have more than three linear factors.
Therefore, f (y) = k (y – 1) (y - 2) (y + 3)
y3-7y+ 6 = k (y – 1) (y - 2) (y + 3)
Putting x = 0 on both sides, we get
0 – 0 + 6 = k (0 – 1) (0 - 2) (0 + 3)
6 = 6k
k = 1
Putting k = 1 in f (y) = k (y – 1) (y - 2) (y + 3), we get
f (y) = (y – 1) (y - 2) (y + 3)
Hence,
y3-7y+ 6 = (y – 1) (y - 2) (y + 3)