Question

Divide \[3{x^2} - {x^3} - 3x + 5\] by \[x - 1 - {x^2}\]

Answer 1

Hint: Here we use the method of long division of polynomials where a polynomial is divided by another polynomial like normal division method. Arrange the dividend and divisor in standard form where the terms are written in descending order of power of variable.
The term that is being divided is called a dividend, the term that divides the term is called divisor, the value that we obtain on dividing is called quotient and the term that remains in the end is called remainder. We divide a by b giving us quotient q and remainder r, we write \[a = bq + r\]
* Long division method: when dividing \[a{x^n} + b{x^{n - 1}} + ....c\] by \[px + q\] we perform as
\[px + q)\overline {a{x^n} + b{x^{n - 1}} + ....c} ((a/p){x^{n - 1}} + ...\]
            \[\underline { - a{x^n} + (qa/p){x^{n - 1}}} \]
                          \[0.{x^n} + (b - qa/p){x^{n - 1}}\]
Here we multiply the divisor with such a term that gives us the exact same term as the highest power in the dividend and then we proceed in the same way. We multiply the divisor with such a factor so we cancel out the highest power of the variable in it.

Complete step-by-step solution:
We are given the dividend as \[3{x^2} - {x^3} - 3x + 5\] and the divisor as \[x - 1 - {x^2}\]
We arrange dividend and divisor in standard form
Then dividend is \[ - {x^3} + 3{x^2} - 3x + 5\]and divisor is\[ - {x^2} + x - 1\]
Now we use the long division method to divide the dividend by divisor.
\[ - {x^2} + x - 1)\overline { - {x^3} + 3{x^2} - 3x + 5} (x - 2\]
                      \[\underline { - {x^3} + {x^2} - x} \]
                                \[2{x^2} - 2x + 5\]
                                \[\underline {2{x^2} - 2x + 2} \]
                                                     \[3\]
When we divide a by b and we get q as quotient and r as remainder we write the equation
\[a = bq + r\]

Here we can write the equation as \[3{x^2} - {x^3} - 3x + 5 = (x - 1 - {x^2})(x - 2) + 3\]

Note: Students are likely to make mistakes while performing the long division method, always keep in mind that sign needs to be changed from negative to positive and vice versa inside the division when we are solving for the next value to be divided by the dividend.
Students should know the process of long division which is just like basic division just involving polynomial equations and the sign change which is the most important part.