341 3.2 Synthetic Division Consider the result shown here. 3x3 - 2x2 - 150 x2 - 4 = 3x - 2 + 12x - 158 x2 - 4 . We can express it using the preceding division algorithm. 3x2 + 10x + 40 x - 4)3x3 - 2x2 + 0x - 150 3x3 - 12x2 10x2 + 0x 10x2 - 40x 40x - 150 40x - 160 10 310 40 -4)3 -2 0 -150 3 -12 10 0 10 - 40 40 -150 40 -160 10 The numbers in color that are repetitions of the numbers directly above them can also be omitted, as shown on the left below. 310 40 -4)3 -2 0 -150 -12 10 0 -40 40 -150 -160 10 310 40 -4)3 -2 0 -150 -12 10 -40 40 -160 10 The numbers in color are again repetitions of those directly above them. They may be omitted, as shown on the right above. The entire process can now be condensed vertically. The rest of the bottom row is obtained by subtracting -12, -40, and -160 from the corresponding terms above them. To simplify the arithmetic, we replace subtraction in the second row by addition and compensate by changing the -4 at the upper left to its additive inverse, 4. 3x3 - 2x2 - 150 = 1x2 - 4213x - 22 + 12x - 158 (+++)+++* (+)+* (+)+* (++)++* ƒ1x g1x2 q1x2 r1x2 Dividend = Divisor # Quotient + Remainder (original polynomial) Additive inverse 4)3 -2 0 -150 12 40 160 Signs changed 3 10 40 10 Quotient 3x2 + 10x + 40 + 10 x - 4 Remainder -4)3 -2 0 -150 -12 -40 -160 3 10 40 10 The top row of numbers can be omitted since it duplicates the bottom row if the 3 is brought down. This is the long-division process for the result shown to the right. 3x - 2 x2 + 0x - 4)3x3 - 2x2 + 0x - 150 3x3 + 0x2 - 12x -2x2 + 12x - 150 -2x2 - 0x + 8 12x - 158 Synthetic Division When a given polynomial in x is divided by a firstdegree binomial of the form x - k, a shortcut method called synthetic division may be used. The example on the left below is simplified by omitting all variables and writing only coefficients, with 0 used to represent the coefficient of any missing terms. The coefficient of x in the divisor is always 1 in these divisions, so it too can be omitted. These omissions simplify the problem, as shown on the right.
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