Question

The polynomial 3x² + mx? - nx - 10 has a factor of (x - 1). When divided by x + 2, the remainder is 36. What are

the values of m and n?

Answer 1

`m = 12`

`n =3`

Explanation:

Given

`P(x) = x^3 + mx^2 - nx - 10`

Required

The values of m and n

For x - 1;

we have:

`x - 1 = 0`

`x=1`

So:

`P(1) = (1)^3 + m*(1)^2 - n*(1) - 10`

`P(1) = 1 + m*1 - n*1 - 10`

`P(1) = 1 + m - n - 10`

Collect like terms

`P(1) = m - n + 1 - 10`

`P(1) = m - n -9`

Because x - 1 divides the polynomial, then P(1) = 0;

So, we have:

`m - n -9 = 0`

Add 9 to both sides

`m - n = 9` --- (1)

For x + 2;

we have:

`x + 2 = 0`

`x = -2`

So:

`P(-2) = (-2)^3 + m*(-2)^2 - n*(-2) - 10`

`P(-2) = -8 + 4m + 2n - 10`

Collect like terms

`P(-2) = 4m + 2n - 10 - 8`

`P(-2) = 4m + 2n - 18`

x + 2 leaves a remainder of 36, means that P(-2) = 36;

So, we have:

`4m + 2n - 18 = 36`

Collect like terms

`4m + 2n = 36+18`

`4m + 2n = 54`

Divide through by 2

`2m + n=27` --- (2)

Add (1) and (2)

`m + 2m - n + n = 9 +27`

`3m =36`

Divide by 3

`m = 12`

Substitute
`m = 12` in (1)

`m - n =9`

Make n the subject

`n = m - 9`

`n = 12 - 9`

`n =3`