Question

Using factor theorem, factorize each of the following polynomial:

x⁴ - 7x³ + 9x² + 7x - 10

Answer 1

Let, f (x) = x⁴ - 7x³ + 9x² + 7x - 10

The constant term in f (x) is equal to -10 and factors of -10 are , +1, +2, +5 and +10

Putting x = 1 in f (x), we have

f (1) = (1)4 – 7 (1)3 + 9 (1)2 + 7 (1) - 10

= 1 – 7 + 9 + 7 - 10

= 0

Therefore, 

(x - 1) is a factor of f (x).

Similarly, (x + 1), (x - 2) and (x - 5) are the factors of f (x).

Since, f (x) is a polynomial of degree 4. So, it cannot have more than four linear factors.

Therefore, 

f (x) = k (x – 1) (x + 1) (x - 2) (x - 5)

x⁴ - 7x³ + 9x² + 7x - 10 = k (x – 1) (x + 1) (x - 2) (x - 5)

Putting x = 0 on both sides, we get

0 + 0 – 0 - 10 = k (0 – 1) (0 + 1) (0 - 2) (0 - 5)

-10 = -10k

k = 1

Putting k = 1 in f (x) = k (x – 1) (x + 1) (x - 2) (x - 5), we get

f (x) = (x – 1) (x + 1) (x - 2) (x - 5)

Hence,

x⁴ - 7x³ + 9x² + 7x - 10 = (x – 1) (x + 1) (x - 2) (x - 5)