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Polynomial Division and Factorization

This is a collection of polynomial division and factorization problems, providing step-by-step instructions and solutions. The problems involve synthetic division, finding factors, and determining remainders.

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Polynomial Division and Factorization

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  1. D M S B 2x2 – 3x – 2 3x + 2 6x3 - 5x2 – 12x – 4 6x3 + 4x2 -9x2 – 12x -9x2 – 6x – 6x – 4 – 6x – 4 0 Check: (3x + 2)(2x2 – 3x – 2) = 6x3 – 9x2 – 6x + 4x2 – 6x – 4 6x3 – 5x2 – 12x – 4

  2. 7 (2x – 1) 2x2 – 3x + 1 – 2x – 1 4x3 – 8x2 + 5x – 8 4x3 – 2x2 -6x2 + 5x -6x2 + 3x 2x – 8 2x – 1 remainder - 7 Check: (2x – 1)(2x2 – 3x + 1) = 4x3 – 6x2 + 2x – 2x2 + 3x – 1 = 4x3 – 8x2 + 5x – 1 Add remainder -7 = 4x3 – 8x2 + 5x – 8

  3. x2 + 3x 5x4 + 18x3 + 10x2 +3x + 9x2 3x3 + 3x + 1 5x2 5x4 + 15x3 3x3 + 10x2 + 3x x2 x2 + 3x 0

  4. x2 + 3x One factor is + 3x + 1 5x2 Factor: 5x4 + 18x3 + 10x2 +3x Divide to find that the other factor is:

  5. Synthetic Division 3 2 1 (3x3 + 8x2 + 5x – 7)  (x + 2) 3 8 5 -7 -2 -6 -4 -2 3 2 1 -9 remainder 9 x + 2 – + x2 + x answer

  6. Synthetic Division (2x4 – 6x3 + x2 – 3x – 3)  (x – 3) 2 -6 1 -3 -3 3 6 0 0 3 -3 2 0 1 0 remainder 2x3 answer + 0x2 + 1x + 0 3 (x – 3) 2x3 + x –

  7. Synthetic Division (x4 – 2x3 – 70x + 20)  (x – 5) 1 -2 0 -70 20 5 5 25 15 75 45 1 3 15 5 remainder 45 (x – 5) answer x3 + 3x2 + 15x + 5 +

  8. Synthetic Division 1 2 (4x3 – 8x2 + 3x – 8)  (2x – 1) 3 2 2 -4 -4 -3 2 1 0 -4 2 -3 0 remainder 4 (2x – 1) answer 2x2 – 3x –

  9. Synthetic Division 1 3 -1 3 -7 3 -2 3 1 3 ( 6x2 – x – 7)  (3x + 1) 2 2 -1 -2 remainder answer 2 (3x + 1) 2x – 1 –

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