Factor and Remainder theorem

1.  Remainder theorem

If a polynomial f(x) is divided by (x – α) then remainder = f(α)

Example: Find the remainder (Without division) on dividing f(x) by (x + 3) where f(x) = 2x2 – 7x – 1

Solution:-

X + 3 = 0    or,  x = -3

put x = -3 in f(x)

f(-3) = 2(-3)(-3) – (-3) – 1 = 20

Hence, remainder is 20.

2. Factor theorem

(x- α) is a factor of the polynomial f(x) if and only if f(α) = 0

Example: Show that (x – 5) is factors of f(x) where f(x) = 2x2 – 11x + 5.

Solution:-

x – 5 = 0   or, x = 5

put x = 5 in f(x)

f(5) = 2(5)(5) – 11(5) + 5 = 50 – 50 = 0

Hence (x – 5) is the factor.

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