Synthetic division - step by step
Here's how the process of synthetic division works, step-by-step.
Divide \(3{x^3} - 4x + 5\) by \((x + 2)\) and state the quotient and remainder.
First, make sure the polynomial is listed in order of descending powers.
Missing powers must be replaced by a zero. \(3{x^3} - 4x + 5\) has coefficients \(3\), \(0\), \(- 4\) and \(5\). The \(0\) is there because there's no \({x^2}\).
Another way of writing \((x + 2)\) is \(x = - 2\). So \(- 2\) is the divisor.
Step one
First write the question in this form:
Step two
Bring down the first coefficient (in this example \(3\)) and write it below the line. Multiply it by the divisor (\(- 2\)) and place the product (\(- 6\)) below the next coefficient but above the line.
Step three
Now add the product you have just calculated (in our example \(- 6\)) to the coefficient above it, (\(0\)). Write the resulting number (\(- 6\)) below the line. Multiply this new number by the divisor (\(- 2\)) and place the answer, (\(12\)) below the next coefficient.
Step four
Continue like this until no more values remain.
The first three numbers below the line are the coefficients of the quotient and the last number is the remainder.
Now that we have the coefficients of the quotient, we write its expression by reducing the original degree by one.
So for our example, the answer is:
\(3{x^3} - 4x + 5\)
\(= (x + 2)(3{x^2} - 6x + 8) - 11\)
where \((x + 2)\) is the divisor, \((3{x^2} - 6x + 8)\) is the quotient and \(- 11\) is the remainder.
If you see that your answer has a common factor in the quotient, then you can simplify. Bring the factor to the front of the brackets and multiply the divisor - remember that the order is not important for multiplication.
