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Chapter 5: Introduction to Polynomials and Polynomial Functions

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Presentation on theme: "Chapter 5: Introduction to Polynomials and Polynomial Functions"— Presentation transcript:

1 Chapter 5: Introduction to Polynomials and Polynomial Functions
11/11/2018 Chapter 5: Introduction to Polynomials and Polynomial Functions 5.1 Introduction to Polynomials and Polynomial Functions 5.2 Multiplication of Polynomials 11/11/2018 Math KM KRM Spring 2009

2 4.1 Introduction to Polynomials and Polynomial Functions
11/11/2018 Math KM

3 Polynomial Example: -3x+10
-3x+10 is a binomial because it has two terms. -3x+10 is the same as -3x1+10x0 is the leading term It is “linear”. The degree is 1. The leading coefficient is -3. is the “constant” term The degree is 0. The coefficient is 10. 5.1 11/11/2018 Math KM

4 Polynomial Example: 5x2 – 3x + 10
5x2 – 3x + 10 is a trinomial because it has three terms. is the leading term It is “quadratic”. The degree is 2. The leading coefficient is 5. is the “linear” term. The degree is 1. The coefficient is -3. is the “constant” term. The degree is 0. The coefficient is 10. 5.1 11/11/2018 Math KM

5 Polynomial Example: 7x3 + 5x2 – 3x + 10
7x3 + 5x2 – 3x + 10 is a polynomial. is the leading term. It is “cubic”. The degree is 3. The leading coefficient is 7. is the “quadratic” term. The degree is 2. The coefficient is 5. is the “linear” term. The degree is 1. The coefficient is -3. is the “constant” term. The degree is 0. The coefficient is 10. 5.1 11/11/2018 Math KM

6 Simplifying a Polynomial
combine "like" terms quadratic terms linear terms constant terms write in Standard Form 5.1 11/11/2018 Math KM

7 Simplify Another Polynomial
combine "like" terms quadratic terms linear terms constant terms write in Standard Form rewrite without the "missing" term (coefficient is zero) 5.1 11/11/2018 Math KM

8 How to EVALUATE a Polynomial Function
Replace the variable with parentheses and substitute. 5.1 11/11/2018 Math KM

9 EVALUATE -5x2 + 3x – 1 for x = 2 5.1 11/11/2018 Math KM

10 EVALUATE -5x2 + 3x – 1 for x = 0 5.1 11/11/2018 Math KM

11 EVALUATE -5x2 + 3x – 1 for x = -2 5.1 11/11/2018 Math KM

12 Adding Polynomials Example 1
combine "like" terms The final answer is x2 - 2x + 4. 5.1 11/11/2018 Math KM

13 Adding Polynomials Example 2
combine "like" terms The final answer is -x2 - 8. 5.1 11/11/2018 Math KM

14 Adding Polynomials Example 3
combine "like" terms Final Answer: 9x3 - 8x2 - 2x + 4 5.1 11/11/2018 Math KM

15 The OPPOSITE of a POLYNOMIAL
can be written two ways: -1( ) = 5.1 11/11/2018 Math KM

16 Subtracting Polynomials: Example 1
change to ADD the OPPOSITE ADD like terms Answer: -9x3 - 6x2 - 8x + 18 5.1 11/11/2018 Math KM

17 Subtracting Polynomials: Example 2
change to ADD the OPPOSITE ADD like terms Answer: -18x2 + x - 5 5.1 11/11/2018 Math KM

18 Subtracting Polynomials: Example 3
change to ADD the OPPOSITE ADD like terms Final Answer: -2x2 + 6x 5.1 11/11/2018 Math KM

19 Subtracting Polynomials: Example 4
change to ADD the OPPOSITE ADD like terms Opposites add to ZERO! Answer: 0 5.1 11/11/2018 Math KM

20 4.2 Multiplication of Polynomials
DISTRIBUTIVE FOIL 11/11/2018 Math KM

21 Multiplying Monomials
Multiply the Coefficients Multiply the Variables using the Product Rule 5.2 4.4 S KM & PP

22 Multiplying Monomials:
Multiply the Coefficients Multiply the Variables Final Answer 5.2 4.4 S KM & PP

23 The Distributive Property: Example 1
The 5 will be multiplied first with x and then with 3. Final Answer: 5x+15 5.2 11/11/2018 Math KM

24 The Distributive Property: Example 2
The -2x will be multiplied first with x and then with 4. Final Answer 5.2 11/11/2018 Math KM

25 The Distributive Property: Example 3
The 3x will be multiplied first with 2x and then with 7. 5.2 11/11/2018 Math KM

26 The Distributive Property: Monomial times Trinomial
The 5x will be multiplied with: x2, 2x and 7. 5.2 11/11/2018 Math KM

27 The Distributive Property: Monomial times Polynomial
The 2x will be multiplied with: 6x3,-4x2, 5x and -3. 5.2 11/11/2018 Math KM

28 Outside Terms Inside Terms First Terms Last Terms F O I L
FOIL: Example 1 Outside Terms Inside Terms First Terms Last Terms F O I L 5.2 11/11/2018 Math KM

29 Foil Example 1: Generic Rectangle
5.2 11/11/2018 Math KM

30 FOIL: Example 2 O I L F 5.2 11/11/2018 Math KM

31 FOIL: Example 3 O I L F 5.2 11/11/2018 Math KM

32 Example 4 Try this one! F O I L 5.2 11/11/2018 Math KM

33 Remember the Distributive Property!
Multiply each term of the binomial to each term of the trinomial Combine Like Terms Write in Standard Form 5.2 11/11/2018 Math KM

34 x+8 and x-8 are CONJUGATES Let’s find their product!
Opposites add to ZERO! Difference of Squares 5.2 11/11/2018 Math KM

35 Let’s Look at some Examples!
Binomial Conjugate 5.2 11/11/2018 Math KM

36 Will there be more conjugates in Algebra ?
Yes. Have a look at a couple examples that come up in more advanced sections: Expression Conjugate 5.2 11/11/2018 Math KM

37 The Shortcut! Multiply Conjugates Opposites add to ZERO!
Difference of Squares 5.2 11/11/2018 Math KM

38 Difference of Squares The Shortcut: Example 1 5.2 11/11/2018
Math KM

39 Difference of Squares The Shortcut: Example 2 5.2 11/11/2018
Math KM

40 Difference of Squares The Shortcut: Example 3 4.2 11/11/2018
Math KM

41 Difference of Squares The Shortcut: Example 4 5.2 11/11/2018
Math KM

42 Difference of Squares The Shortcut: Example 5 5.2 11/11/2018
Math KM

43 What about a Binomial Squared?
F O I L double square square 5.2 11/11/2018 Math KM

44 Shortcut to Square (a+b)
Binomial Squared square the first term minus twice the product of the terms plus the square of the second term double square square Perfect Square Trinomial 5.2 11/11/2018 Math KM

45 Squaring A Binomial: Shortcut Example 1
Binomial Squared double square square Perfect Square Trinomial 5.2 11/11/2018 Math KM

46 Squaring A Binomial: Shortcut Example 2
Binomial Squared double square square Perfect Square Trinomial 5.2 11/11/2018 Math KM

47 Squaring A Binomial: Shortcut Example 3
Binomial Squared double square square Perfect Square Trinomial 5.2 11/11/2018 Math KM

48 Squaring A Binomial: Shortcut Example 4
Binomial Squared double square square Perfect Square Trinomial 5.2 11/11/2018 Math KM

49 Squaring A Binomial: Shortcut Example 5
Binomial Squared double square square Perfect Square Trinomial 5.2 11/11/2018 Math KM

50 Squaring A Binomial: Shortcut Example 6
Binomial Squared double square square Perfect Square Trinomial 5.2 11/11/2018 Math KM

51 That’s All For Now! 11/11/2018 Math KM

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