what is synthatic division??
Answers
Answer:
Synthetic division is a shorthand method to find the quotient and remainder when dividing a polynomial by a monic linear binomial ((a polynomial of the form x-k).x−k).
Step-by-step explanation:
Here's the general procedure; we'll follow this with another example.
(Step 1) Write the coefficients of the polynomial as written in standard form, in order. With the example x^2 + 2x + 6 ,x
2
+2x+6, the coefficients are 1, 2, and 6. If there is a "missing term," then one or more 0s must be used. For example, 5x^4 + 2x^2 - 5 5x
4
+2x
2
−5 has the coefficients 5, 0, 2, 0, and -5. \big((Formally, a "missing term" means the polynomial can be written with at least one term 0x^k, 0x
k
, where kk is a non-negative integer less than the degree of the polynomial.\big))
(Step 2) Draw a vertical line and bar, as shown below.
(Step 3) Given that we are dividing by x-k ,x−k, to the left, write k.k. ((Notice the subtraction; it means if we are dividing by x+1,x+1, i.e. k = -1.)k=−1.)
(Step 4) We start on the far left of the polynomial coefficients and "add" the first coefficient to the number below it—since there's no number, we're "adding" 0, and then write the same number below the line.
(Step 5) We take our result and multiply it by the value kk to the left. We then right result diagonally up and to the right from the last position.
(Step 6) We continue like step 4 and add the next column (this time there are two numbers to add).
(Step 7) We continue like step 5, taking the result of our sum, multiplying by k,k, and writing the result diagonally up and to the right.
(Step 8) The steps 6 and 7 continue until the last column is reached. The final number written will be our remainder. ((If the remainder is 0, that means x-kx−k was a factor of our original polynomial.)) The other numbers on the bottom row represent the result of multiplication; they are the coefficients of the quotient's polynomial, and the degree of the quotient's polynomial is 1 less than the degree of the original polynomial.
In order to divide polynomials using synthetic division, you must be dividing by a linear expression and the leading coefficient (first number) must be a 1. For example, you can use synthetic division to divide by x + 3 or x – 6, but you cannot use synthetic division to divide by x2 + 2 or 3x2 – x + 7.