synthetic department is a shortcut technique for dividing a polynomial with the aid of linear factors in which the leading coefficient is 1. If it's not identical to 1, then it requires enhancing the dividend to make the main coefficient 1 before the use of the technique. This approach provides you with the quotient (the end result of the department) and the remainder. It uses the divisors of the form "(x + a)" and "(x - a)".
\(\frac{P(x)}{(x-a)} =\ Q(x) + \frac{R}{(x-a)}\)
Where:
solve the given values beneath by using the use of the artificial division approach.
Navigate with the steps to evaluate polynomials using the synthetic division method:
\(\dfrac{4 x^{3} - 6 x^{2} + 2 x - 8}{x - 3}\)
Step #1: Write The Coefficients Of The Dividend
4, -6, 2, -8
Step #2: Write Zeros Of Linear Factors As The Divisor
x - 3 = 0
x = 3
Step #3: Write values in synthetic division format:
\(\begin{array}{c|rrrrr}& x^{3} & x^{2} & x^{1} & x^{0} \\ 3.0 & 4 & -6 & 2 & -8 \\ & & \\ \hline & \end{array}\)
Step #4: Carry Down The Leading Coefficient In The Next Column
\(\begin{array}{c|rrrrr}3.0 & 4 & -6 & 2 & -8 \\ & & \\ \hline & 4 \end{array}\)
Step #5: Multiply The Leading Coefficient By The Divisor
Using the synthetic division to find zeros, multiply the obtained value by the denominator and write the result into the next column.
\( 4 \times (3.0) = 12\)
Write The Outcome In The Next Column
\(\begin{array}{c|rrrrr}3.0 & 4 & -6 & 2 & -8 \\ & & 12 & \\ \hline & 4 & \end{array}\)
Step #6: Repeat Steps 4 & 5 Until The Last Coefficient
\(\begin{array}{c|rrrrr}3.0 & 4 & -6 & 2 & -8 \\ & & 12 & 18 & 60 \\ \hline & 4 & 6 & 20 & 52 \end{array}\)
Step #7: Value in Last Column is Remainder, The Number From Right Is Quotient
The quotient is \( 4 x^{2} + 6 x + 20\), and the remainder is \(52\)
Therefore, the final answer is:
\(\dfrac{4 x^{3} - 6 x^{2} + 2 x - 8}{x - 3} = 4 x^{2} + 6 x + 20 + \dfrac{52}{x - 3}\)
This division method is applicable whilst the divisor of a polynomial is a linear issue in the shape ax + b, in which the highest energy of x is 1. For this department in case you want whole calculations, take assist from this synthetic department solver that simplifies the division with given steps.
No, the synthetic department method most effective works to divide polynomials via linear expressions (a binomial of the shape x-c, wherein c is represented as a steady).
The calculator simplest works to divide polynomials the usage of synthetic department as we already said above.
