SECTION A.4 Synthetic Division A33 in row 4 is the remainder. Since row 1 is not really needed, we can compress the process to three rows, where the bottom row contains both the coefficients of the quotient and the remainder. ) x 3 2 1 0 3 6 15 45 2 5 15 48 − − − − − Recall that the entries in row 3 are obtained by subtracting the entries in row 2 from those in row 1. Rather than subtracting the entries in row 2, we can change the sign of each entry and add. With this modification, our display will look like this: ) x 3 2 1 0 3 6 15 45 2 5 15 48 − − Notice that the entries in row 2 are three times the prior entries in row 3. Our last modification to the display replaces the −x 3 by 3. The entries in row 3 give the quotient and the remainder, as shown next. ) − + + x x 3 2 1 0 3 6 15 45 2 5 15 48 2 5 15 48 2 Quotient Let’s go through an example step by step. Using Synthetic Division to Find the Quotient and Remainder Use synthetic division to find the quotient and remainder when − − − x x x 4 5 is divided by 3 3 2 Solution EXAMPLE 1 Step 1 Write the dividend in descending powers of x. Then copy the coefficients, remembering to insert a 0 for any missing powers of x. 1 4 0 5 − − Row 1 Step 2 Insert the usual division symbol. In synthetic division, the divisor is of the form −x c, and c is the number placed to the left of the division symbol. Here, since the divisor is −x 3, insert 3 to the left of the division symbol. )3 1 4 0 5 − − Row 1 Step 3 Bring the 1 down two rows, and enter it in row 3. )3 1 4 0 5 1 − − ↓ Step 4 Multiply the latest entry in row 3 by 3, and place the result in row 2, one column over to the right. )3 1 4 0 5 3 1 − − Step 5 Add the entry in row 2 to the entry above it in row 1, and enter the sum in row 3. )3 1 4 0 5 3 1 1 − − − Row 1 Row 2 (subtract) Row 3 Row 1 Row 2 (add) Row 3 Row 1 Row 2 (add) Row 3 Remainder 3 × 3 × 3 × Row 1 Row 2 Row 3 Row 1 Row 2 Row 3 Row 1 Row 2 Row 3 3 × 3 × (continued)
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