Question

How do you long divide `2y^3 - 5y^2+ 22y` by `2y+3`?

Answer 1

Long dividing polynomials is similar to long dividing integers. See explanation for details.

Result: `y^2-4y+17` with remainder `-51`

Explanation:

Please excuse the poor formatting, but it's the easiest way to get the symbols to line up.

Write down the polynomial dividend under the bar in standard form with descending powers of the variable (in this case `y`). If any powers are missing, write down a term with `0` coefficient. In our case the constant term is missing, so I have written a `+0` term.

Write down the polynomial divisor outside the bar to the left. Also write it in standard form, adding any missing powers with a `0` coefficient.

Write the first term of the quotient above the bar, choosing it to match the highest order term of the dividend when multiplied by the highest order term of the divisor. In our case, we write down `y^2` since `2y*y^2 = 2y^3`

Write down the result of multiplying the first term of the quotient by the divisor below the dividend.

Subtract to get a remainder (in our case `-8y^2`) and bring down the next term from the dividend alongside it (resulting in `-8y^2+22y`).

Write down the next term `-4y` of the quotient, chosen so that when multiplied by the divisor, the highest order term matches the highest order term `-8y^2` of our running remainder.

Write down the result (`-8y^2-12y`) of multiplying this second term of the quotient by the divisor below our running remainder.

Keep going in similar fashion until we're left with a remainder `-51` that is of lower degree than the divisor.