![O
POLYNOMIAL FUNCTIONS
EVALUATING A POLYNOMIAL FUNCTION
Find 2g(-2a) if g(x) = 3x2
– 2x – 6
2g(-2a) = 2[3(-2a)2
– 2(-2a) – 6]
2g(-2a) = 2[12a2
+ 4a – 6]
2g(-2a) = 24a2
+ 8a – 12](https://image.slidesharecdn.com/lecture2apolynomialfunctionsfactorremaindertheorem-250725113041-320e1c66/85/Lecture-2A_Polynomial-Functions-Factor_Remainder-Theorem-pptx-8-320.jpg)
Lecture 2A_Polynomial Functions Factor_Remainder Theorem.pptx
- 1. MATH 121 ALGEBRA ....Linking Learners Everywhere © 2013 All Rights Reserved SAMUEL N. N. NORTEY LECTURE 2B: POLYNOMIAL FUNCTIONS
- 2. DEFINITION: What is a Polynomial? POLYNOMIAL FUNCTIONS ....Linking Learners Everywhere © 2013 All Rights Reserved • Polynomial: Is a mathematical expression consisting of variables, coefficients and exponents, combined using addition, subtraction and multiplication operations. • GENERAL FORM: 1 2 1 2 1 0 ( ) ... n n n n P x a x a x a x a x a
- 3. DEFINITION: What is a Polynomial? POLYNOMIAL FUNCTIONS ....Linking Learners Everywhere © 2013 All Rights Reserved • Polynomial: is a monomial or a sum of monomials.
- 4. EVALUATING POLYNOMIAL ....Linking Learners Everywhere © 2013 All Rights Reserved • EVALUATING POLYNOMIAL FUNCTIONS:. Find f(-2) if f(x) = 3x2 – 2x – 6 f(-2) = 3(-2)2 – 2(-2) – 6 f(-2) = 12 + 4 – 6 f(-2) = 10
- 5. POLYNOMIAL FUNCTIONS ....Linking Learners Everywhere © 2013 All Rights Reserved EVALUATING A POLYNOMIAL FUNCTION Find f(-2) if f(x) = 3x2 – 2x – 6 f(-2) = 3(-2)2 – 2(-2) – 6 f(-2) = 12 + 4 – 6 f(-2) = 10
- 6. ....Linking Learners Everywhere 2013 All Rig hts Reserved POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find f(2a) if f(x) = 3x2 – 2x – 6 f(2a) = 3(2a)2 – 2(2a) – 6 f(2a) = 12a2 – 4a – 6
- 7. ....Linking Learners Everywhere 2013 All Rig hts Reserved POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find f(m + 2) if f(x) = 3x2 – 2x – 6 f(m + 2) = 3(m + 2)2 – 2(m + 2) – 6 f(m + 2) = 3(m2 + 4m + 4) – 2(m + 2) – 6 f(m + 2) = 3m2 + 12m + 12 – 2m – 4 – 6 f(m + 2) = 3m2 + 10m + 2
- 8. O POLYNOMIAL FUNCTIONS EVALUATING A POLYNOMIAL FUNCTION Find 2g(-2a) if g(x) = 3x2 – 2x – 6 2g(-2a) = 2[3(-2a)2 – 2(-2a) – 6] 2g(-2a) = 2[12a2 + 4a – 6] 2g(-2a) = 24a2 + 8a – 12
- 9. POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1 Max. Zeros: 1
- 10. POLYMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2 Max. Zeros: 2
- 11. POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3 Max. Zeros: 3
- 12. POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function Degree = 4 Max. Zeros: 4
- 13. POLYNOMIAL FUNCTIONS GENERAL SHAPES OF POLYNOMIAL FUNCTIONS f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1 Quintic Function Degree = 5 Max. Zeros: 5
- 14. POLYNOMIAL FUNCTIONS END BEHAVIOR Degree: Even Leading Coefficient: + End Behavior: As x -∞; f(x) +∞ As x +∞; f(x) +∞ f(x) = x2
- 15. POLYNOMIAL FUNCTIONS END BEHAVIOR Degree: Even Leading Coefficient: – End Behavior: As x -∞; f(x) -∞ As x +∞; f(x) -∞ f(x) = -x2
- 16. POLYNOMIAL FUNCTIONS END BEHAVIOR Degree: Odd Leading Coefficient: + End Behavior: As x -∞; f(x) -∞ As x +∞; f(x) +∞ f(x) = x3
- 17. POLYNOMIAL FUNCTIONS END BEHAVIOR Degree: Odd Leading Coefficient: – End Behavior: As x -∞; f(x) +∞ As x +∞; f(x) -∞ f(x) = -x3
- 18. REMAINDER AND FACTOR THEOREMS f(x) = 2x2 – 3x + 4 2 -3 4 2 4 2 Divide the polynomial by x – 2 Find f(2) f(2) = 2(2)2 – 3(2) + 4 f(2) = 8 – 6 + 4 6 f(2) = 6 1 2
- 19. REMAINDER AND FACTOR THEOREMS f(x) = 3x5 – 4x3 + 5x - 3 Find f(-3) Try this one: Remember – Some terms are missing When synthetic division is used to evaluate a function, it is called SYNTHETIC SUBSTITUTION.
- 20. REMAINDER AND FACTOR THEOREMS ....Linking Learners Everywhere © 2013 All Rights Reserved The binomial x – a is a factor of the polynomial f(x) if and only if f(a) = 0. FACTOR THEOREM
- 21. REMAINDER AND FACTOR THEOREMS Is x – 2 a factor of x3 – 3x2 – 4x + 12 2 1 -3 -4 12 1 2 -1 -2 -6 0 Yes, it is a factor, since f(2) = 0. Can you find the two remaining factors?
- 22. REMAINDER AND FACTOR THEOREMS (x + 3)( ? )( ? ) = x3 – x2 – 17x - 15 Find the two unknown ( ? ) quantities.
- 23. SUMMARY LECTURE 2A • EVALUATING POLYNOMIALS • GENERAL SHAPE OF POLYNOMIAL FUNCTIONS • END BEHAVIOR • REMAINDER AND FACTOR THEOREM ....Linking Learners Everywhere © 2013 All Rights Reserved
- 24. Contact ....Linking Learners Everywhere © 2013 All Rights Reserved Phone: Email: snortey@gctu.edu.gh Skype: snorteygtuc Website: gtuc.edu.gh