When x  2 is divided into f x, the remainder is 3. Determine the remainder when x  2 is

Question

x  2, the remainder is – 7. What are the values of m and g?
f x is divided by Determine the remainder when x3  3x2  x  2 is divided by x  3x  5.
divided into each of the following:
fx1 b) fxx2 c) fx4x7 d) 2fx7 e) fx2

Answer 1

To determine the remainder when x+2 is divided by f(x), we can write the division as:

x+2 = q(x) * f(x) + 3

where q(x) is the quotient.

To determine the remainder when x+2 is divided by x+2, we can write the division as:

x+2 = q(x) * (x+2) - 7

where q(x) is the quotient.

From the given information, we can set up the following equations:

x+2 = q(x) * f(x) + 3
x+2 = q(x) * (x+2) - 7

Solving these equations simultaneously will give us the values of m and g:

q(x) * f(x) + 3 = q(x) * (x+2) - 7
q(x) * f(x) - q(x) * (x+2) = -10
q(x) (f(x) - (x+2)) = -10
q(x) = -10 / (f(x) - (x+2))

Now, we need more information about f(x) to determine the values of m and g.

To determine the remainder when (x^3 + 3x^2 - x - 2) is divided by (x+3)(x+5), we can use polynomial long division or synthetic division. Since polynomial long division can be lengthy to type, I will use synthetic division:

We set up the division:

-3 -5
---------------
3 | 1 3 -1 -2
-3 0 3
---------------
1 0 2 1

So, the remainder when (x^3 + 3x^2 - x - 2) is divided by (x+3)(x+5) is 2x + 1.

Finally, we can evaluate the expressions:

a) f(x)+1: The remainder is 3 + 1 = 4.
b) f(x)+x+2: The remainder is (2x + 1) + x + 2 = 3x + 3.
c) f(x)+(4x+7): The remainder is (2x + 1) + (4x + 7) = 6x + 8.
d) 2f(x)-7: The remainder is 2(2x + 1) - 7 = 4x - 5.
e) [f(x)]^2: The remainder is (2x + 1)^2 = 4x^2 + 4x + 1.