Question

How do you divide `(x ^ { 3} - 4x ^ { 2} + 5x - k ) \div ( x - 3)`?

Answer 1

Quotient `=x^2-x+2` , remainder `6-k`

Explanation:

Rewrite `x^3−4x^2+5x−k` in the form

`x^3−4x^2+5x−k = x^3-3x^2-x^2+3x+2x-6+6-k`
`= x^2(x-3)-x(x-3)+2(x-3)+6-k`
`=(x-3)(x^2-x+2)+(6-k)`

Answer 2

`x^2-x+2+(6-k)/(x-3)`

Explanation:

`"one way is to use the divisor as a factor in the numerator"`

`"consider the numerator"`

`color(red)(x^2)(x-3)color(magenta)(+3x^2)-4x^2+5x-k`

`=color(red)(x^2)(x-3)color(red)(-x)(x-3)color(magenta)(-3x)+5x-k`

`=color(red)(x^2)(x-3)color(red)(-x)(x-3)color(red)(+2)(x-3)color(magenta)(+6)-k`

`"quotient "=color(red)(x^2-x+2)," remainder "=6-k`

`rArr(x^3-4x^2+5x-k)/(x-3)`

`=x^2-x+2+(6-k)/(x-3)`