Week 8 The remainder Theorem.pptx

THE REMAINDER THEOREM R E V WHO, I WHERE AND E WHEN W FOUND IT? MECHANICS: The students will group into 5. Students will solve an expression given in board. The students will find the answer on the given screen with the corresponding n...

THE REMAINDER THEOREM R E V WHO, I WHERE AND E WHEN W FOUND IT? MECHANICS: The students will group into 5. Students will solve an expression given in board. The students will find the answer on the given screen with the corresponding names. Student will answer the questions in 5mins. MOTIVATION “WHO REMAINS?” MECHANICS: The students will group into 5. Students will find who left out on the other groups. Students will only be given a minutes to identify who remains. example – group 1 will group the group 2 who they choice to be remain to their groups. The group that who give the mystery, will give the correct answer to teacher before the other group will guess.  OBJECTIVES State The Find the Solve word Remainder remainder problems using Theorem. when a the Remainder polynomial is Theorem. divided by a binomial using The Remainder Theorem. The remainder is determined using two methods -long division and the synthetic division. The remainder is the quantity left after a number or expression can no longer be divided exactly by another number or expression. The remainder is 0 if the number or expression can be divided exactly by the divisor.  EXAMPLE: For example, LONG DIVISION 2 2𝑥 +𝑥−1 𝑥 − 4) 3 2 −(2 𝑥 − 8 𝑥 ) 2 𝑥 −5 𝑥 −(𝑥 ¿¿ 2 −4 𝑥)¿ − 𝑥+ 4 −(− 𝑥 +4) remainder  EXAMPLE: For example, SYNTHETIC DEVISION 2 − 7− 5 4 4 8 4 − 4 remainder To find the remainder when the polynomials p(x) is divided by x - c, simply evaluate p(x) for x = c. in other words, simply find p(c). This is the essence of the Remainder Theorem. The Remainder Theorem if a polynomial p(x) is divided by x - c , then the remainder is equal to p(c). To find the remainder when the polynomials p(x) is divided by x - c, simply evaluate p(x) for x = c. in other words, simply find p(c). This is the essence of the Remainder Theorem. The Remainder Theorem if a polynomial p(x) is divided by x - c , then the remainder is equal to p(c).  EXAMPLE: Determine the remainder when the given polynomial is divided by the given divisor a. b.  PROOF OF THE REMAINDER THEOREM In any division problem, the relation Holds among these quantities. So if and represent the quotient and remainder respectively, when the dividend is divided by the divisor  PROOF OF THE REMAINDER THEOREM 𝑝 ( 𝑥) 𝑥−𝑐 =𝑞 ( 𝑥 )+ 𝑅 𝑥−𝑐 𝑒 𝑞𝑢𝑎𝑡𝑖𝑜𝑛 1 Multiply both side of equation 1 by , results in𝑝 ( 𝑥 )=𝑞 ( 𝑥 ) ∙ ( 𝑥 − 𝑐 ) + 𝑅 𝑒 𝑞𝑢𝑎𝑡𝑖𝑜𝑛 2 Evaluate of equation 2 for 𝑝 ( 𝑐 ) =𝑞 ( 𝑐 ) ∙ ( 𝑐 − 𝑐 ) + 𝑅 𝑝 ( 𝑐 ) =𝑞 ( 𝑐 ) ∙ 0+ 𝑅 𝑝 ( 𝑐)=𝑅 𝑒 𝑞𝑢𝑎𝑡𝑖𝑜𝑛 3 Equation 3 confirms that the remainder is equal to  ACTIVITY Answer the activity on Mental Math A. pp. 80 Number 1-5