Find the remainder when f(x) = 2x³ - 12x² + 11x + 2 is divided by x - 5.
3, - 7, 7, - 3

Solution:

If p(x) be any polynomial of degree greater than or equal to 1.

Suppose that when p(x) is divided by x - a, the quotient is q(x) and the remainder is r(x),

i.e.

p(x) = (x - a)q(x) + r(x)

Since the degree of x - a is 1 and the degree of r(x) is less than the degree of x - a, the degree of r(x) = 0.

This means that r(x) is a constant, say r.

Therefore for every value of x,

r(x) = r.

Hence,

p(x) = (x - a) q(x) + r

In particular,

if x = a the equation gives us

p(a) = (a - a)q(a) + r = r

f(x) = 2x³ - 12x² + 11x + 2, and the zero of x - 5 is 5

On substituting 5 in the given polynomial, we get

f(5) = 2(5)³ - 12(5)² + 11(5) + 2

= 250 - 300 + 55 + 2

= 7

Therefore, the remainder when f(x) = 2x³ - 12x² + 11x + 2 is divided by x - 5 is 7.

---

Find the remainder when f(x) = 2x³ - 12x² + 11x + 2 is divided by x - 5.
3, - 7, 7, - 3

Summary:

Using the remainder theorem the remainder after dividing f(x) = 2x³ - 12x² + 11x + 2  by x - 5 is 7.