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Understanding Long Division, Synthetic Division, and the Remainder Theorem in Polynomials

This guide explores the methods of long division and synthetic division for polynomials, as well as the Remainder Theorem. It details how to determine if a number is a root of a polynomial and demonstrates these concepts through various examples, including the proper methods for subtracting by adding the opposite. You'll learn about the procedures for dividing factors and confirming if values like -4 and -1 are roots based on the remainder produced. This comprehensive approach helps in effectively solving polynomial equations.

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Understanding Long Division, Synthetic Division, and the Remainder Theorem in Polynomials

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Presentation Transcript

  1. Notes Day 6.3 Can only divide by x-a Long Division Synthetic Division Remainder Theorem

  2. Long Division: Divide Factor x2 - 1 + 0 Subtract by adding the opposite 0 - 2x 0 + 0 X2-1 -2x + 3 • -2x + 3 Is 3/2 a root? 0 Yes because there was no remainder!

  3. Long Division: Divide Factor - 5 x + Subtract by adding the opposite -5x - 2 -5x - 20 Is -4 a root? 18 No because there was a remainder!

  4. Synthetic Division: Divide Root 2 3 -4 15 -3 9 -6 -15 0 5 -3 2 Is -3 a root? 2X2-3x+5 Yes because there was no remainder!

  5. Synthetic Division: Divide Root 3 -4 2 -1 -1 7 -3 -9 -10 9 -7 3 3X2-7x+9+ Is -1 a root? No because there was a remainder!

  6. Remainder Theorem -4 1 0 -5 4 12 • 160 16 -4 -44 172 -40 11 -4 1 Find f(-4) Notice f(-4) = the remainder above!!!!!

  7. Remainder Theorem Find P(-1) -1 2 6 -5 0 -61 • -9 -4 -2 9 -70 9 -9 4 2 You can find P(-1) using synth div or substitution

  8. Add To Notes! Not factorable Divide by -1 using synthetic division -1 1 2 -11 -12 Add -1 -1 12 0 -12 1 1 Multiply Factor Now If you can’t factor then QF or compl the square!

  9. Add To Notes! Not factorable Divide by 1 using synthetic division 1 1 4 -3 -2 Add 5 1 2 0 2 5 1 Multiply You can’t factor so QF or complete the square!

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