Question

How do you find the quotient `(3q^2+20q+11)div(q+6)` using long division?

Answer 1

This is the same as long division. The only difference to the traditional method is the format.

`3q+2-1/(q+6)`

Explanation:

`" "3q^2+20q+11`
`color(magenta)(3q)(q+6)->ul(3q^2 +18q) larr" Subtract"`
`" "color(white)(.)0+color(white)(2)2q+11`
`color(magenta)(color(white)(2)2)(q+6)->" "ul(2q+12 ) larr" Subtract"`
`" "0color(magenta)(-1 larr" Remainder")`

`color(magenta)(3q+2-1/(q+6))`

Answer 2

Please see the explanation.

Explanation:

Given:

`color(white)( (q + 6)/color(black)(q + 6))color(white)((d + e + f))/("|" color(white)(x)3q^2 + 20q + 11)`

Find the first term in the quotient by dividing first term in the dividend by the first term in the divisor:

`(3q^2)/q = 3q`

Write the first term in the quotient:

`color(white)( (q + 6)/color(black)(q + 6))(3qcolor(white)(d+ e + f))/("|" color(white)(x)3q^2 + 20q + 11)`

Multiply that term by the divisor `3q(q+6) = 3q^2+18q` subtract this from the dividend:

`color(white)( (q + 6)/color(black)(q + 6))(3qcolor(white)(d+ e + f))/("|" color(white)(x)3q^2 + 20q + 11)`
`color(white)("..........")ul(-3q^2-18q`
`color(white)(".........................")2q+ 11`

Find the next term in the quotient by dividing the first non-zero term in the results of the subtraction by the first term in the divisor:

`(2q)/q = 2`

Write this term into the quotient:

`color(white)( (q + 6)/color(black)(q + 6))(3q+2color(white)(e. + f))/("|" color(white)(x)3q^2 + 20q + 11)`
`color(white)("..........")ul(-3q^2-18q`
`color(white)(".........................")2q+ 11`

Multiply that term by the divisor `2(q+6) = 2q+12` subtract this from the dividend:

`color(white)( (q + 6)/color(black)(q + 6))(3q+2color(white)(e. + f))/("|" color(white)(x)3q^2 + 20q + 11)`
`color(white)("..........")ul(-3q^2-18q`
`color(white)(".........................")2q+ 11`
`color(white)("......................")ul(-2q-12`
`color(white)("................................")-1`

Because the power of the divisor is greater than the power of the results of the subtraction, we know that we are done and that `-1` is a remainder.

The quotient can be expressed as:

`3q+2-1/(q+6)`