Question

How do you long divide $f\left( x \right) = {x^3} + 3{x^2} - 2x + 1$ by $\left( {x + 1} \right)$?

Answer 1

Hint:The given cubic polynomial $f\left( x \right) = {x^3} + 3{x^2} - 2x + 1$ is divided by linear polynomial $\left( {x + 1} \right)$ so we get the quotient as a quadratic polynomial. It is given to divide the polynomial by a long division method so our basic aim is to cancel the highest power term in the first step then the second-highest power term in the second step proceed continuously until the order of remainder is less than the order of divisor. In this case, the order of the remainder will be zero.

Complete step by step answer:
Here, the given cubic polynomial is $f\left( x \right) = {x^3} + 3{x^2} - 2x + 1$. It is to be divided by a linear polynomial $\left( {x + 1} \right)$ which is called a divisor.
Steps of long division:
Step 1: First multiply the divisor by a suitable number such that the product's highest power term is equal to the highest term of given polynomials.
Step 2: Subtract the product from the given polynomial by changing the sign of product terms as shown below.
Step 3: Then bring down the next term and apply step 1 to cancel the second-highest power terms.
Step 4: Repeat the first two steps until the order of remainder is less than the order of the divisor.
Now,
$\begin{array}{l}
\left. {x + 1} \right){x^3} + 3{x^2} - 2x + 1\left( {{x^2} + 2x - 4} \right.\\
\,\,\,\,\,\,\,\,\,\,\,\,\underline {{x^3} + {x^2}} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2{x^2} - 2x\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline {2{x^2} + 2x} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 4x + 1\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { - 4x - 4} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5
\end{array}$
Hence, $\left( {{x^2} + 3{x^2} - 2x + 1} \right) = \left( {x + 1} \right)\left( {{x^2} + 2x - 4} \right) + 5$.

Note:
This question is also solved by the factor method. To divide by factor method, we have to first factorize the given polynomial and then divide the factors by divisor and we get that divisor cancels one of the factors of the polynomial and the remaining factors will be the quotient for the division of the given polynomials. This method is applicable only for the polynomials which are completely divisible by the divisor polynomial.