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

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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 Divide x 4 – 10x 2 – 2x + 4 by x

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Long Division

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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 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) = 2x 4 + 7x 3 – 4x 2 – 27x –

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Example 6 continued Show that (x – 2) and (x + 3) are factors of f(x) = 2x 4 + 7x 3 – 4x 2 – 27x –

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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. 1.r = f(k) 2.If r = 0 then (x – k) is a factor of f(x). 3.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 [2 nd ][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 [2 nd ][QUIT] Run SYN Enter 2/3

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