1. Remainder and Factor Theorem (1) Intro to Polynomials -degree -identities -division (long, short, synthetic) (2) Remainder Theorem -finding remainders -special case  Factor Theorem -factorise & solve cubic equations

2. Intro to Polynomials Degree Terms Coefficient Constant Value

3. http://www.youtube.com/watch?v=18OFfTyic7g More detailed Intro to Polynomials Simple Intro to Polynomials http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_05/ brainpops/index.php4/na Intro to Polynomials

4. Long Division of Polynomials http://www.youtube.com/watch?v=l6_ghhd7kwQ http://www.youtube.com/watch?v=FTRDPB1wR5Y Simple Example More difficult example

5.  Example 1: DividendDivisorQuotient In this case, the division is exact and Dividend = Divisor x Quotient Long Division of Polynomials

6.  Example 2: The number 7 when divided by 2 will not give an exact answer. We say that the division is not exact. [7 = (2 x 3) + remainder 1 ] In this case, when the division is NOT exact, Dividend = Divisor x Quotient + Remainder Long Division of Polynomials

7.  Definition of degree: For any algebraic expression, the highest power of the unknown determines the degree.  For division of polynomials, we will stop dividing until the degree of the expression left is smaller than the divisor. Algebraic Expression Degree 2x + 11 x 3 - 5x3 -3x 2 + x + 42

8. Division by a Monomial Divide: Rewrite: Divide each term separately:

9. Division by a Binomial Divide: Divide using long division Insert a place holder for the missing term x 2

10. Division of Polynomials  Division of polynomials is similar to a division sum using numbers. Consider the division 10 ÷ 2 = 5 10 2 5 0 Consider the division ( x 2 + x ) ÷ ( x + 1 ) 0 - -

11. - - 0 - - - Example 1:Example 2:

12.  When the division is not exact, there will be a remainder. Consider the division 7 ÷ 2 7 2 3 6 1 Consider (2x 3 + 2x 2 + x) ÷ (x + 1) - - - remainder

13. - - - Example 1: Degree here is not smaller than divisor’s degree, thus continue dividing Degree here is less than divisor’s degree, thus this is the remainder

14. - - Example 2: Degree here is less than divisor’s degree, thus this is the remainder

15. - - Example 3:

16. ‘Short’ Division of Polynomials Examples

17. Synthetic Division of Polynomials http://www.youtube.com/watch?v=bZoMz1Cy1T4 http://www.youtube.com/watch?v=nefo9cUo-wg http://www.youtube.com/watch?v=4e9ugZCc4rw *http://www.youtube.com/watch?v=1jvjL9DtGC4http://www.youtube.com/watch?v=1jvjL9DtGC4 Preview Example: the link from long division to synthetic division http://www.mindbites.com/lesson/931-int-algebra-synthetic-division- with-polynomials Examples: how to perform synthetic division on linear divisors (and the link to remainder theorem) Extra: how to perform synthetic division on quadratic divisors

18. Remainder and Factor Theorem Introduction to Remainder Theorem http://library.thinkquest.org/C0110248/algebra/remfactintro.htm http://www.youtube.com/watch?v=PJd26kdLxWw

19. Remainder and Factor Theorem Introduction to Factor Theorem http://www.youtube.com/watch?v=WyPXqe-KEm4&feature=related Use of Factor Theorem to solve polynomial equations http://www.youtube.com/watch?v=nXFlAj7zBzo&feature=related http://www.youtube.com/watch?v=tBjSW365pno&feature=related http://www.youtube.com/watch?v=7qcCOry8FoQ&feature=related