Practice Distributed and Integrated \( \begin{array}{ll}\text { Lse synthetic division to divide. } & \text { "2. } 2+4 x-2 x^{2}+4 x^{2} \text { by }(-3+x) \\ \text { 1. } 2 x^{4}-x \text { by }(x-2) & \text { isin }\end{array} \)

Below are two worked‐examples using synthetic division. (Remember: when a polynomial is missing any terms, you must include them with zero coefficients.) ───────────────────────────── Example 1. Divide 2x⁴ – x by x – 2. Step 1. Write the dividend in descending order, inserting any missing terms. Here,   2x⁴ + 0·x³ + 0·x² – 1·x + 0. So the coefficients are: 2, 0, 0, –1, 0. Step 2. For synthetic division with divisor x – 2, use the zero 2. Set up: ______________________________ 2 | 2  0  0   –1  0 | ? ? ? ? Process: a) Bring down the leading coefficient: 2. b) Multiply 2 by 2 → 4; write under the next coefficient. c) Add: 0 + 4 = 4. d) Multiply 4 by 2 → 8; write under the next coefficient. e) Add: 0 + 8 = 8. f) Multiply 8 by 2 → 16; write under the next coefficient. g) Add: (–1) + 16 = 15. h) Multiply 15 by 2 → 30; write under the constant. i) Add: 0 + 30 = 30. The synthetic division table looks like: 2   0   0   –1   0 ---------------------------- 2 | 2   4   8   15  30 Step 3. Interpretation: – The bottom row (except the final number) gives the coefficients of the quotient:   Quotient: 2x³ + 4x² + 8x + 15. – The final number (30) is the remainder. Thus,   (2x⁴ – x) ÷ (x – 2) = 2x³ + 4x² + 8x + 15  with remainder 30, or equivalently,   (2x⁴ – x) = (x – 2)(2x³ + 4x² + 8x + 15) + 30. ───────────────────────────── Example 2. Divide 4x³ – 2x² + 4x + 2 by –3 + x (which can be written as x – 3). Step 1. Write the dividend in descending order. It is already ordered:   4x³ – 2x² + 4x + 2, with coefficients: 4, –2, 4, 2. Step 2. For synthetic division with divisor x – 3, use the zero 3. Set up: ____________________________ 3 | 4  –2  4  2 Process: a) Bring down the leading coefficient: 4. b) Multiply 4 by 3 → 12; write under the next coefficient. c) Add: –2 + 12 = 10. d) Multiply 10 by 3 → 30; write under the next coefficient. e) Add: 4 + 30 = 34. f) Multiply 34 by 3 → 102; write under the constant. g) Add: 2 + 102 = 104. The table becomes: 4   –2   4   2 ------------------------- 3 | 4   10   34  104 Step 3. Interpretation: – The quotient is given by the numbers (except the last): 4x² + 10x + 34. – The remainder is 104. Thus,   (4x³ – 2x² + 4x + 2) ÷ (x – 3) = 4x² + 10x + 34  with remainder 104, or in equation form,   4x³ – 2x² + 4x + 2 = (x – 3)(4x² + 10x + 34) + 104. ───────────────────────────── Summary of Answers: 1. (2x⁴ – x) ÷ (x – 2) = 2x³ + 4x² + 8x + 15 remainder 30. 2. (4x³ – 2x² + 4x + 2) ÷ (x – 3) = 4x² + 10x + 34 remainder 104. These examples illustrate how to use synthetic division for dividing polynomials.