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Remainder and Factor Theorem

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. Terms. Degree. Coefficient. Constant. Value.

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Remainder and Factor Theorem

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  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. Terms Degree Coefficient Constant Value Intro to Polynomials

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

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

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

  6. Long Division of Polynomials • 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

  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.

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

  9. Insert a place holder for the missing term x2 Division by a Binomial Divide: Divide using long division

  10. Division of Polynomials • Division of polynomials is similar to a division sum using numbers. Consider the division 10 ÷ 2 = 5 Consider the division ( x2 + x ) ÷ ( x + 1 ) 5 2 10 - 10 - 0 0

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

  12. Consider the division 7 ÷ 2 Consider (2x3 + 2x2 + x) ÷ (x + 1) • When the division is not exact, there will be a remainder. 3 2 7 - - 6 1 - remainder -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 theremainder

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

  15. Example 3: - -

  16. ‘Short’ Division of Polynomials Examples

  17. Synthetic Division of Polynomials 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) 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=1jvjL9DtGC4 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

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