Factor Theorem & Rational Root Theorem Lesson PDF

LPO4: Conscientious, Adept PERFORMERS & ACHIEVERS I am a conscientious, adept performer and achiever, competently pursuing my mission in life. EPO4: Search voluntarily beyond readily available sources of information, resources, and standard techniques to generate new under...

LPO4: Conscientious, Adept PERFORMERS & ACHIEVERS I am a conscientious, adept performer and achiever, competently pursuing my mission in life. EPO4: Search voluntarily beyond readily available sources of information, resources, and standard techniques to generate new understandings towards workable solutions to Factor Theorem & Rational Root Theorem Prepared by: Miss Romalyn DC. Villegas  OBJECTIVES At the end of the lesson, you should be able to search voluntarily beyond readily available sources of information, resources, and standard techniques to generate new understandings towards workable solutions to existing problems:  Proving the Factor Theorem and Rational Root Theorem.  Consider this!is divided by When The remainder obtained is 0. By division, we have: 𝒇 ( 𝒙 ) =( 𝒙 +𝟐 ). 𝑸 ( 𝒙 ) +𝟎 𝒇 ( 𝒙 ) =( 𝒙 +𝟐 ). 𝑸 ( 𝒙 ) This means that is a factor of that is is divided by. For a polynomial then, is a factor of. FACTOR Conversely , if is a factor of a THEOREM polynomial , then  FACTOR THEOREM Example: 1. Determine whether is a factor of each polynomial. a b Solutio (Evaluate 𝒙 − 𝟒=𝟎 Solutio directly; n: substitution) 3 2 𝒙 =𝟒n: (Evaluate substitution) directly; 𝑃 ( 𝑥 ) =𝑥 −2 𝑥 −11 𝑥+12 ; 𝑃 ( 4) 𝟑 𝟐 ; 𝑃 ( 4) 𝑃 ( 𝑥 ) =𝟐 𝒙 +𝒙 −𝟐 𝒙 −𝟏𝟓 𝑃 ( 4 () =¿ 4− 3 2 ) 2 ( 4−11 (4) ) +12 𝑃 ( 4 ) =¿ ) ( 4− 2 ( 4+ 3 ) 2(− 2 4)15 ¿ 64 − 44 − 32+12 ¿ 128 +16−−815 𝑃 (4)=0 𝑃 ( 4 ) =121 Since the remainder is 0, Since the remainder is 121 (not Therefore is a factor of 0), Therefore is NOT a factor of  FACTOR THEOREM Example: 2. Find the value of k for which the binomial is a factor of Solutio n: 𝑃 ( 𝑥 ) =0 (𝑈𝑠𝑒 𝐹𝑎𝑐𝑡𝑜𝑟 𝑇h𝑒𝑜𝑟𝑒𝑚) 𝑃 ( − 4 ) =0 (𝐺𝑖𝑣𝑒𝑛: 𝑥+4=0 → 𝑥 =−4 ; 𝑅=0) 4 3 ( − 4 ) +𝑘( − 4 ) −4 (− 4)2 ¿0 (𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒) 256+𝑘(−64) −4(16) ¿ 0 ( 𝑆𝑖𝑚𝑝𝑙𝑖𝑓𝑦 ) 256−64𝑘−64¿ 0 4 3 2 −64𝑘 ¿−256 +64 𝑇h𝑢𝑠 , 𝑡h𝑒𝑏𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑥+4𝑖𝑠 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 𝑥 −𝟑 𝑥 −4 𝑥. −64𝑘 −192 𝒌=𝟑  RATIONAL ROOT THEOREM The possible rational roots are formed as , where p is a factor of a constant term and q is a factor of the leading term. Examples : 1. Find the possible rational roots of the equation Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ±1 ±1  RATIONAL ROOT THEOREM The possible rational roots are formed as , where p is a factor of a constant term and q is a factor of the leading term. Examples : 1. Find the possible rational roots of the equation Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ± 1± 2  RATIONAL ROOT THEOREM The possible rational roots are formed as , where p is a factor of a constant term and q is a factor of the leading term. Examples : 1. Find the possible rational roots of the equation Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1± 2 ± 1± 2± 1  RATIONAL ROOT THEOREM The possible rational roots are formed as , where p is a factor of a constant term and q is a factor of the leading term. Examples : 1. Find the possible rational roots of the equation Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1± 2 ±2 ± 1± 2± 1 ±2  RATIONAL ROOT THEOREM The possible rational roots are formed as , where p is a factor of a constant term and q is a factor of the leading term. Examples : 1. Find the possible rational roots of the equation Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1± 2 ±2±4 ± 1± 2± 1 ±2±1  RATIONAL ROOT THEOREM The possible rational roots are formed as , where p is a factor of a constant term and q is a factor of the leading term. Examples : 1. Find the possible rational roots of the equation Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1± 2 ±2±4 ±4 ± 1± 2± 1 ±2±1 ±2  RATIONAL ROOT THEOREM The possible rational roots are formed as , where p is a factor of a constant term and q is a factor of the leading term. Examples : 1. Find the possible rational roots of the equation Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1± 2 ±2±4 ±4 ± 1± 2± 1 ±2±1 ±2 Possible Rational 1  RATIONAL ROOT THEOREM The possible rational roots are formed as , where p is a factor of a constant term and q is a factor of the leading term. Examples : 1. Find the possible rational roots of the equation Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1± 2 ±2±4 ±4 ± 1± 2± 1 ±2±1 ±2 Possible Rational± 1 ,  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ±1 ±1  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ± 1± 3  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3 ± 1± 3 ±1  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3±3 ± 1± 3 ±1±3  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3± 3± 5 ± 1± 3 ±1± 3± 1  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3± 3± 5 ±5 ± 1± 3 ±1± 3± 1 ±3  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3± 3± 5 ±5±9 ± 1± 3 ±1± 3± 1 ±3±1  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3± 3± 5 ±5± 9± 9 ± 1± 3 ±1± 3± 1 ±3± 1± 3  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3± 3± 5 ±5± 9± 9 ±15 ± 1± 3 ±1± 3± 1 ±3± 1± 3 ±1  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3± 3± 5 ±5± 9± 9 ±15±15 ± 1± 3 ±1± 3± 1 ±3± 1± 3 ± 1± 3  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3± 3± 5 ±5± 9± 9 ±15±15±45 ± 1± 3 ±1± 3± 1 ±3± 1± 3 ± 1± 3± 1  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3± 3± 5 ±5± 9± 9 ±15±15±45±45 ± 1± 3 ±1± 3± 1 ±3± 1± 3 ± 1± 3± 1± 3 Possible 1 Rational Roots: ±1,± 3 ±3, , ±1,±5, ±9,±3,±15,± 5,±45,± 15,  RATIONAL ROOT THEOREM Examples : List all the possible rational zeroes of the equation 2. Solutio n:  List all the factors of (constant term) and (leading coefficient) Factors of : →𝑝 Factors of : →𝑞  Divide each factor of p by ± 1± 1 ±3± 3± 5 ±5± 9± 9 ±15±15±45±45 ± 1± 3 ±1± 3± 1 ±3± 1± 3 ± 1± 3± 1± 3 , Possible Rational Roots: Activity A. #3 Write YES if the polynomial is a factor of the given binomial and NO, if not. 1) 2) 3) 4) 5) Activity # A. 4 List all the possible zeroes of each polynomial. 1) 2) 3) 4) 5)