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### Factor Theorem and Remainder Theorem

• 1. F a C Ro t Theorem R e M a i N d e R THEOREM
• 2. REMAINDER THEOREM: If the polynomial P(X) is divided by X-C, then the remainder is P(C). FACTOR THEOREM: If the remainder comes out to be 0 (zero), then X-C is a factor of P(X).
• 4. f(x)= x4 – 13x2 + 36 𝑝 𝑞 = ±1,±2,±3,±4,±6,±9,±12,±18,±36 ±1 Possible roots: ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36 1.
• 5. If x= 1: f(1) = (1)4 – 13(1)2 + 36 = 1 – 13(1) + 36 = 1 – 13 + 36 = 24 Therefore, x=1 is not a root and (x-1) is not a factor. If x= -1: f(-1) = (-1)4 – 13(-1)2 + 36 = 1 – 13(1) + 36 = 1 – 13 + 36 = 24 Therefore, x=-1 is not a root and (x+1) is not a factor.
• 6. If x= 2: f(2) = (2)4 – 13(2)2 + 36 = 16 – 13(4) + 36 = 16 – 52 + 36 = 0 Therefore, x=2 is a root and (x-2) is a factor. If x= -2: f(-2) = (-2)4 – 13(-2)2 + 36 = 16 – 13(4) + 36 = 16 – 52 + 36 = 0 Therefore, x=-2 is a root and (x+2) is a factor.
• 7. If x= 3: f(3) = (3)4 – 13(3)2 + 36 = 81 – 13(9) + 36 = 81 – 117 + 36 = 0 Therefore, x=3 is a root and (x-3) is a factor. If x= -3: f(-3) = (-3)4 – 13(-3)2 + 36 = 81 – 13(9) + 36 = 81 – 117 + 36 = 0 Therefore, x=3 is a root and (x-3) is a factor.
• 8. Since the exponent of the polynomial function is 4, there should be four roots and four factors. The roots of the polynomial function, f(x) = x4 – 13x2 + 36, are ± 2 𝑎𝑛𝑑 ± 3 . The factors of the polynomial function are (x-2) (x+2) (x-3) (x+3).
• 9. f(x) = x2– 12x + 9 𝑝 𝑞 = ±1,±3,±9, ±1 if x = 1 = 12- 10(1) + 9 = 1 -10 + 9 =0 Therefore, x=1 is a root and (x-1) is a factor. 2.
• 10. If x = -1 = 12- 10(-1) + 9 = 1 +10 + 9 =20 Therefore, x=-1 is a not a root and (x+1) is not a factor. if x = 3 = 32- 10(3) + 9 = 9 - 30 + 9 = -2 Therefore, x=3 is a not root and (x-3) is not a factor.
• 11. if x = -3 = -32- 10(-3) + 9 = 9 + 30 + 9 = 48 Therefore, x=-3 is not a root and (x+3) is a not factor. if x = 9 = 92- 10(9) + 9 = 81 -90 + 9 = 0 Therefore, x=9 is a root and (x-9) is a factor.
• 12. If x =- 9 = 92- 10(9) + 9 = -81 + 90 + 9 = 18 Therefore, x=-9 is not a root and (x+9) is not a factor. Since the exponent of the polynomial function is 2, there should be two roots and two factors. The roots of the polynomial function, f(x) = x2– 12x + 9 ate +1 𝑎𝑛𝑑 + 9. The factors of the polynomial function are (x-1) (x-9)
• 13. Danielle Erika L. Diva Ronalie C. Mejos Mark Lenon F. Dacir Rafael C. Vallejos MEMBERS X- Einstein
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