9. When
4
3 2
x mx gx is divided by
x 2
, the remainder is – 2. When it is divided by
x 1
, the remainder is – 3. Find m and g.
To find the values of m and g, we can use the remainder theorem.
According to the remainder theorem, if a polynomial f(x) is divided by x - a, then the remainder is equal to f(a).
In this case, when f(x) = 3x^2 + 2x + g, the remainder when divided by x + 2 is -2, so:
f(-2) = -2
Substituting -2 for x in the polynomial equation, we get:
3(-2)^2 + 2(-2) + g = -2
12 - 4 + g = -2
8 + g = -2
g = -10
Similarly, when f(x) is divided by x + 1, the remainder is -3, so:
f(-1) = -3
Substituting -1 for x in the polynomial equation, we get:
3(-1)^2 + 2(-1) - 10 = -3
3 - 2 - 10 = -3
-9 = -3
Therefore, the values of m and g are m = -9 and g = -10.