how to divide polynomial synthetics division?
Elsewhere, we introduced the concept of synthetic division. Now we are going to amplify the concept and develop it more fully. The basic idea is to take a "big" (higher order) polynomial and divide it into a "small" (lower order) polynomial. If it divides evenly, we have in effect partially factored the polynomial.
For example, let’s divide the following expression:
We’ll take the example step-by-step. The larger polynomial is the dividend, the smaller is called the divisor. First, set up the division:
Looking at the first term in the dividend, we ask, "How many times does the first term of the divisor (x) need to be multiplied to get the first term of the dividend (x4)? The answer is x3. Write x3 on top of the bar above the x4 term, multiply both terms of the divisor by x3, and write each result below the dividend term with the same power of x. Subtract:
Notice that the x4 terms cancel, and that –3x3 – (–5x3) = 2x3. For the next step, we bring down the next term (-7x2), so the new dividend term is 2x3 - 7x2. Again examining the highest order terms, we notice that 2x3 / x = 2x2, so our next term is 2x2: copyright ybarra-cgm
Now, bring down the next term (-11x) and do the next division 3x2 / x = 3x:
Last step: 4x / x = 4, so we put a 4 on top and perform the final division step:
The remainder is zero, so we have found that x = 5 is a zero of the polynomial and a root of the equation.
When you actually do these problems, you will do a single calculation that looks like the last figure above. If the dividend is not evenly divided by the divisor, the space where the 0 goes at the end will be occupied by a number R, the remainder (see an example of a remainder problem later). The above process greatly resembles ordinary long division, and is in fact a generalization of ordinary long division, where x is always 10.
The above process can be streamlined. Compare the following to the above.
Here’s how the process works. The first line is the divisor (just the coefficient of x0 = 1) inside the angle bracket, followed by the dividend, stripped of all powers of x, but written in order of decreasing powers of x. Missing powers of x must be represented by a zero in that place. The second line is the working line, where intermediate calculations are written one at a time. The bottom line is the quotient. The places in the quotient represent powers of x "one fewer" than the corresponding entry in the dividend directly above.
ybarra-cgm.com
The first coefficient is just brought down to the third line, corresponding to the operation x4 / x = x3. The divisor number in brackets (the 5) then multiplies this number. The result (5) is written below the next place on the second line. Now, the numbers in this column are added (-3 + 5) and the result (2) is written below on the third line. Multiply this new result (2) by the divisor (5) and write the new intermediate result (10) in the next available space on the second line. Add again (-7 + 10) and write the result (3) below on the third line. Multiply this new result (3) by the divisor (5) and write the intermediate result (15) in the next available slot on the second line. Add again (-11 + 15) and write the result (4) below on the third line. Multiply this last result (4) by the divisor (5), write the result (20) below in the last slot on the second line.
The how to divide polynomial synthetics division? information post by website user , webSITE not guarantee correctness ,
More Questions and Answers...