Question

When the polynomial mx^3 - 3x^2 +nx +2 is divided by x+3, the remainder is -4. When it is divided by x-2, the remainder is -4. Determine the value of m and n.

Answer 1

`\begin{gathered} m\text{ =-2} \\ n\text{ =11} \end{gathered}`

Step-by-step explanation:

Here, we want to find the value of m and n

If we substituted a supposed root into the parent polynomial, the value after evaluation is the remainder. If the remainder is zero, then the value substituted is a root.

for x+ 3

x + 3 = 0

x = -3

Substitute this into the first equation as follows:

`\begin{gathered} m(-3)^3-3(-3)^2-3(n)+\text{ 2 = -4} \\ -27m\text{ -27-3n+ 2 = -4} \\ -27m\text{ -3n = -4}+27-2 \\ -27m-3n\text{ = 21} \\ -9m\text{ - n = 7} \end{gathered}`

We do this for the second value as follows:

x-2 = 0

x = 2

Substitute this value into the polynomial:

`\begin{gathered} m(2)^3-3(2)^2+2(n)\text{ + 2 = -4} \\ 8m\text{ - 12 +2n + 2 = -4} \\ 8m\text{ + 2n = -4-2+12} \\ 8m\text{ + 2n = 6} \\ 4m\text{ + n = 3} \end{gathered}`

Now, we have two equations so solve simultaneously:

`\begin{gathered} -9m-n\text{ = 7} \\ 4m\text{ + n = 3} \end{gathered}`

Add both equations:

`\begin{gathered} -5m\text{ = 10} \\ m\text{ =-}(10)/(5) \\ m\text{ = -2} \end{gathered}`

To get the value of n, we simply susbstitute the value of m into any of the two equations. Let us use the second one:

`\begin{gathered} 4m\text{ +n = 3} \\ 4(-2)\text{ + n = 3} \\ -8\text{ + n = 3} \\ n\text{ = 8 + 3} \\ n\text{ = 11} \end{gathered}`