# Factor Theorem-7

### From Homeworkwiki

**Without actual division prove that x ^{4} + 2x^{3} -2x^{2} + 2x - 3 is exactly divisible**

**by x**

^{2}+ 2x - 3**Solution:** Let f(x) = x^{4} + 2x^{3} -2x^{2} + 2x - 3

g(x) = x^{2} + 2x - 3

= x(x + 3) - 1(x + 3) = (x - 1) (x + 3)

Now f(x) will be exactly divisible by g(x) if it is exactly divisible by (x - 1) as well as (x + 3)

i.e. if f(1) = 0 and f ( -3) = 0

Now f(1) = 1^{4} + 2.1^{3} -2.1^{2} + 2.1 - 3

= 1 + 2 - 2 + 2 - 3 = 0

=> (x - 1) is a factor of f(x)

f ( -3) = (-3)^{4} + 2.(-3)^{3} -2.(-3)^{2} + 2.(-3) - 3

= 81 - 54 - 18 - 6 - 3 = 0

=> (x + 3) is a factor of f(x).

=> (x - 1) (x + 3) divides f (x) exactly

Therefore, x^{2} + 2x - 3 is a factor of f(x)