A34 APPENDIX Review Step 6 Repeat Steps 4 and 5 until no more entries are available in row 1. )3 1 4 0 5 3 3 9 1 1 3 14 − − − − − − − Step 7 The final entry in row 3, the −14, is the remainder; the other entries in row 3, the 1, −1, and −3, are the coefficients (in descending order) of a polynomial whose degree is 1 less than that of the dividend. This is the quotient. That is, = − − = − x x Quotient 3 Remainder 14 2 Check: ⋅ + Divisor Quotient Remainder ( ) ( ) ( ) ( ) ( ) = − − − + − = − − − + + + − = − − = x x x x x x x x x x 3 3 14 3 3 3 9 14 4 5 Dividend 2 3 2 2 3 2 Row 1 Row 2 Row 3 3 × 3 × 3 × Using Synthetic Division to Verify a Factor Use synthetic division to show that +x 3 is a factor of + − + − + x x x x x 2 5 2 2 2 3 5 4 3 2 Solution EXAMPLE 2 The divisor is ( ) + = − − x x 3 3 , so place −3 to the left of the division symbol. Then the row 3 entries will be multiplied by −3, entered in row 2, and added to row 1. )32 5 2 2 2 3 6 3 3 3 3 2 1 1 1 1 0 − − − − − − − − Because the remainder is 0, we have ⋅ + Divisor Quotient Remainder ( ) ( ) =+ −+−+= + − + −+ x x x x x x x x x x 3 2 1 2 5 2 2 2 3 4 3 2 5 4 3 2 As we see, +x 3 is a factor of + − + − + x x x x x 2 5 2 2 2 3. 5 4 3 2 Row 1 Row 2 Row 3 As Example 2 illustrates, the remainder after division gives information about whether the divisor is, or is not, a factor. We say more about this in Chapter 4. Now Work problems 9 and 19 Let’s do an example in which all seven steps are combined. ) A.4 Assess Your Understanding Concepts and Vocabulary 1. To check division, the expression that is being divided, the dividend, should equal the product of the and the plus the . 2. To divide − + x x 2 5 1 3 by +x 3 using synthetic division, the first step is to write . 3. Multiple Choice Choose the division problem that cannot be done using synthetic division. (a) − + − x x x 2 4 6 8 3 2 is divided by −x 8 (b) − x 3 4 is divided by +x 1 (c) + − + x x x 3 9 2 5 2 is divided by +x 10 (d) − + − + x x x x 5 3 9 13 4 3 2 is divided by + x 5 2 4. Multiple Choice Choose the correct conclusion based on the following synthetic division: )5 2 3 38 15 10 35 15 2 7 3 0 − − − − − − (a) +x 5 is a factor of + − − x x x 2 3 38 15 3 2 (b) −x 5 is a factor of + − − x x x 2 3 38 15 3 2 (c) +x 5 is not a factor of + − − x x x 2 3 38 15 3 2 (d) −x 5 is not a factor of + − − x x x 2 3 38 15 3 2 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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