1. Polynomial & Synthetic Division MATH 109 - Precalculus S. Rook

2. Overview Section 2.3 in the textbook: – Polynomial long division – Synthetic division – Remainder & factor theorems 2

3. Polynomial Long Division

4. Recall that there are four main steps in long division which cycle until the last digit is brought down: – Division – Multiplication – Subtraction – Bring Down Can write the result in the form: n = q · k + r where n is the dividend (number inside), q is the quotient, k is the divisor (number outside), and r is the remainder Applies to polynomial long division – Need to remember to apply the Distributive Property when necessary 4

5. Polynomial Long Division (Continued) Some tips for polynomial division: – Write the polynomial in descending degree – Fill any missing terms with placeholder zeros – When performing the division step, use the highest term of the divisor (polynomial outside) Polynomial division is helpful because in most cases it is easier to examine a lower degree quotient than the original polynomial – i.e. Breaks the original polynomial apart – E.g. (x 3 – 1) / (x – 1) → (x 3 – 1) = (x 2 + x + 1) · (x – 1) + 0 A remainder of 0 indicates that (x – 1) is a factor of (x 3 – 1) which means it divides evenly 5

6. Polynomial Long Division (Example) Ex 1: Use polynomial division: a) b) 6

7. Synthetic Division

8. A faster variant of polynomial long division Can be used ONLY when the divisor is in the form (x – k) – i.e. When the divisor is linear – Polynomial long division can be used for any polynomial divisor Synthetic division has two main steps which cycle: – Multiply – Add Remember to put the dividend into descending degree 8

9. Synthetic Division (Continued) To perform a synthetic division: – Consider just the coefficients of the dividend Synthetic division uses ONLY numbers Insert a 0 for any term that may be missing – If given a divisor in the form of a factor (x – k), convert it to a zero of the form x = k E.g. (x – 5) → x = 5 and (2x + 3) → x = - 3 ⁄ 2 – Bring down the first digit of the dividend After performing the division: – Write the quotient as a polynomial starting with one degree less than the dividend – The last column represents the remainder 9

10. Setting Up a Synthetic Division The following represents how (x 3 – 1) / (x – 1) would be set up as a synthetic division: – What do you notice? The dividend is missing x 2 and x The factor of (x – 1) is converted to the zero x = 1 – What is the quotient written as a polynomial? x 2 + x + 1 10

11. Synthetic Division (Example) Ex 2: Use synthetic division: a) b) 11

12. Remainder & Factor Theorems

13. Remainder Theorem Sometimes we just need to examine the remainder of a division Remainder Theorem: given the polynomial function f(x), the remainder of f(x) / (x – k) is f(k) when (x – k) is linear – i.e. Convert x – k to the zero x = k and evaluate in f(x) to obtain the remainder – E.g. f(x) = x 3 – 1 → remainder of x 3 – 1 / x – 1 = f(1) = 0 – E.g. f(x) = 2x 2 – 3x + 4 → remainder of 2x 2 – 3x + 4 / x + 2 = f(-2) = 18 Can verify via polynomial division or synthetic division 13

14. Factor Theorem Especially important is the case when the remainder is 0 Factor Theorem: (x – k) is a factor of the polynomial function f(x) if f(k) = 0 – i.e. Recall that f(k) = 0 means the remainder is 0 – i.e. (x – k) divides evenly into f(x) 14

15. Remainder & Factor Theorems (Example) Ex 3: a) Use the Factor Theorem to show that x comprises a factor of f(x) b) Use synthetic division to reduce f(x) c) Factor the resulting polynomial completely, listing both the factored form of f(x) and its real zeros f(x) = x 3 – 7x + 6; x = 2 15

16. Summary After studying these slides, you should be able to: – Perform polynomial division – Synthetic division – Apply the Remainder & Factor Theorems Additional Practice – See the list of suggested problems for 2.3 Next lesson – Complex Numbers (Section 2.4) 16