Question

How do you divide `(4p^4-17p^2+14p-3) div (2p-3)`?

Answer 1

`(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3+3p^2-4p+1)`

Explanation:

The way I would divide it would go as follows:

Write down the first term `color(blue)(2p^3)` of the quotient so that when multiplied by the leading term of the divisor `(color(purple)(2p))` gives the leading term of the dividend `4p^4`.

`(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3)...`

Next note that `(color(purple)(-3))*(color(blue)(2p^3)) = -6p^3` whereas what we want is `0p^3`. So the next term of the quotient is `color(blue)(3p^2)`, which will give us `6p^3` when multiplied by `color(purple)(2p)`, cancelling out the `-6p^3`...

`(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3+3p^2)...`

Next note that `(color(purple)(-3))*(color(blue)(3p^2)) = -9p^2` whereas what we want is `-17p^2`. So the next term of the quotient is `color(blue)(-4p)`, which will give us `-8p^2` when multiplied by `color(purple)(2p)`.

`(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3+3p^2-4p)...`

Next note that `(color(purple)(-3))*(color(blue)(-4p)) = 12p` whereas what we want is `14p`. So the next term of the quotient is `color(blue)(1)`, which will give us `2p` when multiplied by `color(purple)(2p)`.

`(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3+3p^2-4p+1)...`

Finally note that `(color(purple)(-3))*(color(blue)(1)) = -3`, which is exactly what we want, so the division is exact, with no remainder.

`(4p^4-17p^2+14p-3) -: (color(purple)(2p-3)) = color(blue)(2p^3+3p^2-4p+1)`