Section 2.3 Polynomial and Synthetic Division

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

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 .

Long Division. Check

Divide.

Long Division. Check

Example = Check

Division is Multiplication

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).

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.

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

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

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

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

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

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.

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

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