Question

How do you long divide `(2x^2 + x – 16) ÷ (x – 3)`?

Answer 1

(see below for detailed long division)
`2x+7color(white)("XX")"Remainder: "5`
or
`2x+7+5/(x-3)`

Explanation:

Write as a long division:
`color(white)("XXxX")underline(color(white)("XXXXXXXXX"))`
`x-3)2x^2+ color(white)("x")x-16`

Divide the first term of the divisor into the first term of the dividend
`color(white)("XXxX")underline(color(red)(2x) color(white)("XXXXXXX")`
`color(blue)(x)-3)color(blue)(2x^2)+ color(white)("x")x-16`

Multiply the quotient term times the entire divisor
`color(white)("XXxX")underline(color(blue)(2x) color(white)("XXXXXXX")`
`color(blue)(x-3))2x^2+ color(white)("x")x-16`
`color(white)("XXxX")underline(color(red)(2x^2-6x))`

Subtract the product from the dividend
`color(white)("XXxX")underline(2x color(white)("XXXXXXX")`
`x-3)color(blue)(2x^2+ color(white)("x")x)-16`
`color(white)("XXxX")underline(color(blue)(2x^2-6x))`
`color(white)("XXXXXXX")color(red)(7x)`

"Bring down" the next term from the dividend
`color(white)("XXxX")underline(2x color(white)("XXXXXXX")`
`x-3)2x^2+ color(white)("x")xcolor(blue)(-16)`
`color(white)("XXxX")underline(2x^2-6x)`
`color(white)("XXXXXXX")7xcolor(red)(-16)`

Divide the first term of the divisor into the first term of the remainder
`color(white)("XXxX")underline(2x + color(white)("X")color(red)(7)color(white)("XXXX")`
`color(blue)(x)-3)2x^2+ color(white)("x")x-16`
`color(white)("XXxX")underline(2x^2-6x)`
`color(white)("XXXXXXX")color(blue)(7x)-16`

Multiply the most recent quotient term times the entire divisor
`color(white)("XXxX")underline(2x +color(white)("X")color(blue)(7)color(white)("XXXX")`
`color(blue)(x-3))2x^2+ color(white)("x")x-16`
`color(white)("XXxX")underline(2x^2-6x)`
`color(white)("XXXXXXX")7x-16`
`color(white)("XXXXXXX")underline(color(red)(7x-21))`

Subtract the product from the most recent remainder
`color(white)("XXxX")underline(2x +color(white)("X")7color(white)("XXXX")`
`x-3)2x^2+ color(white)("x")x-16`
`color(white)("XXxX")underline(2x^2-6x)`
`color(white)("XXXXXXX")color(blue)(7x-16)`
`color(white)("XXXXXXX")underline(color(blue)(7x-21))`
`color(white)("XXXXXXXXxXX")color(red)(5)`

`color(white)("XXxX")underline(color(white)("XXXXXXXX")`
`x-3)2x^2+ color(white)("x")x-16color(white)("XXX")=2x+7color(white)("XX")"Remainder: " 5`#