2What you should learnHow to use long division to divide polynomials by other polynomialsHow to use synthetic division to divide polynomials by binomials of the form(x – k)How to use the Remainder Theorem and the Factor Theorem
31. 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, Add8. Bring down -6.9. x goes into 6x?6 times.10. Multiply (x-1) by 6.11. Change sign, Add .
9The 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 thatWhere r(x) = 0 or the degree of r(x) is less than the degree of d(x).
10Proper and ImproperSince 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.
13The Remainder TheoremIf a polynomial f(x) is divided by x – k, the remainder is r = f(k).
14The 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 off(x) = 2x4 + 7x3 – 4x2 – 27x – 1827-4-27-18+24221836921118
15Show that (x – 2) and (x + 3) are factors of f(x) = 2x4 + 7x3 – 4x2 – 27x – 1827-4-27-18+242218369-321118-6-15-9253Example 6 continued
16Uses 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.
17Fun with SYN and the TI-83 Use SYN program to calculate f(-3) [STAT] > EditEnter 1, 8, 15 into L1, then [2nd][QUIT]Run SYNEnter -3
18Fun with SYN and the TI-83 Use SYN program to calculate f(-2/3) [STAT] > EditEnter 15, 10, -6, 0, 14 into L1, then [2nd][QUIT]Run SYNEnter 2/3