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**Section 2.3 Polynomial and Synthetic Division**

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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

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

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Long Division. Check

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Divide.

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Long Division. Check

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Example = Check

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**Division is Multiplication**

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**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).

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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.

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**Synthetic Division 1 -10 -2 4 -3 -3 +9 -3 3 -1 1 1 1 -3**

Divide x4 – 10x2 – 2x + 4 by x + 3 1 -10 -2 4 -3 -3 +9 -3 3 -1 1 1 1 -3

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

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

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**The Factor Theorem 2 7 -4 -27 -18 +2 4 22 18 36 9 2 11 18**

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 2 11 18

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**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 -3 2 11 18 -6 -15 -9 2 5 3 Example 6 continued

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**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.

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**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

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**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

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