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Polynomial Division Techniques: Long & Synthetic Methods

Learn to divide polynomials using long and synthetic division with step-by-step examples and applications. Understand the Remainder Theorem, Factor Theorem, and practical uses of division remainders. Enhance your understanding of improper and proper rational expressions through these polynomial division methods.

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Polynomial Division Techniques: Long & Synthetic Methods

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  1. Section 2.3 Polynomial and Synthetic Division

  2. What you should learn • How to use long division to divide polynomials by other polynomials • How to use synthetic division to divide polynomials by binomials of the form (x – k) • How to use the Remainder Theorem and the Factor Theorem

  3. x2 times. 1. x goes into x3? 2. Multiply (x-1) by x2. 3. Change sign, Add. 4. Bring down 4x. 5. x goes into 2x2? 2xtimes. 6. Multiply (x-1) by 2x. 7. Change sign, Add 8. Bring down -6. 9. x goes into 6x? 6times. 10. Multiply (x-1) by 6. 11. Change sign, Add .

  4. Long Division. Check

  5. Divide.

  6. Long Division. Check

  7. Example = Check

  8. Division is Multiplication

  9. The Division Algorithm If f(x) and d(x) are polynomials such that d(x)≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exists a unique polynomials q(x) and r(x) such that Where r(x) = 0 or the degree of r(x) is less than the degree of d(x).

  10. Proper and Improper • Since the degree of f(x) is more than or equal to d(x), the rational expression f(x)/d(x) is improper. • Since the degree of r(x) is less than than d(x), the rational expression r(x)/d(x) is proper.

  11. Synthetic Division Divide x4 – 10x2 – 2x + 4 by x + 3 1 0 -10 -2 4 -3 -3 +9 -3 3 -1 1 1 1 -3

  12. Long Division. 1 -2 -8 3 3 3 -5 1 1

  13. The Remainder Theorem If a polynomial f(x) is divided by x – k, the remainder is r = f(k).

  14. The Factor Theorem A polynomial f(x) has a factor (x – k) if and only if f(k) = 0. Show that (x – 2) and (x + 3) are factors of f(x) = 2x4 + 7x3 – 4x2 – 27x – 18 2 7 -4 -27 -18 +2 4 22 18 36 9 0 2 11 18

  15. Example 6 continued 2 7 -4 -27 -18 +2 Show that (x – 2) and (x + 3) are factors of f(x) = 2x4 + 7x3 – 4x2 – 27x – 18 4 22 18 36 9 -3 2 11 18 -6 -15 -9 0 2 5 3

  16. Uses of the Remainder in Synthetic Division The remainder r, obtained in synthetic division of f(x) by (x – k), provides the following information. • r = f(k) • If r = 0 then (x – k) is a factor of f(x). • If r = 0 then (k, 0) is an x intercept of the graph of f.

  17. Fun with SYN and the TI-83 • Use SYN program to calculate f(-3) • [STAT] > Edit • Enter 1, 8, 15 into L1, then [2nd][QUIT] • Run SYN • Enter -3

  18. Fun with SYN and the TI-83 • Use SYN program to calculate f(-2/3) • [STAT] > Edit • Enter 15, 10, -6, 0, 14 into L1, then [2nd][QUIT] • Run SYN • Enter 2/3

  19. 2.3 Homework • 1-67 odd

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