Remainder and Factor Theorem (1) Intro to Polynomials -degree -identities -division (long, short, synthetic) (2) Remainder Theorem -finding remainders.

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Presentation transcript:

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

Intro to Polynomials Degree Terms Coefficient Constant Value

More detailed Intro to Polynomials Simple Intro to Polynomials brainpops/index.php4/na Intro to Polynomials

Long Division of Polynomials Simple Example More difficult example

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

 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

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

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

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

Example 1:Example 2:

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

- - - 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

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

- - Example 3:

‘Short’ Division of Polynomials Examples

Synthetic Division of Polynomials * Preview Example: the link from long division to 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

Remainder and Factor Theorem Introduction to Remainder Theorem

Remainder and Factor Theorem Introduction to Factor Theorem Use of Factor Theorem to solve polynomial equations