Question

if the polynomial x³+mx²+nx+6 has(x-2) as a factor and leaves 3 as a remainder when divided by x-3 find the value of m and n

Answer 1

your answer in the picture

3m+n=-6

m= -5

so. 3×-5+n=-6

-15+n=-6

n=-6+15

n=9

Answer 2

$\underline\mathfrak\purple{Answer:}$

Let p(x) be the polynomial x³ + mx² - x + 6, then

p(x) =x³ + mx² - x + 6

As (x-2) is a factor of p(x) = x³ + mx² - x + 6, then

p(2) = 0

(2)³ + m(2)² - (2) + 6 = 0

8 + 4m - 2 + 6 = 0

4m + 12 = 0

m = -12 / 4

m = -3

By remainder theorem, we know that

p(x) when divided by x-3 gives a remainder equal to p(3), i.e.,

p(3) = n (given)

(3)³ + m(3)² - 3 + 6 = n

27 + 9m + 3 = n

9m + 30 = n

9(-3) + 30 = n

-27 + 30 = n

n = 3

Hence, the value of m and n are -3 and 3, respectively.