1. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §5.2 Multiply PolyNomials

2. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §5.1 → PolyNomial Functions  Any QUESTIONS About HomeWork §5.1 → HW-15 5.1 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 3 Bruce Mayer, PE Chabot College Mathematics Multiply Monomials  Recall Monomial is a term that is a product of constants and/or variables Examples of monomials: 8, w, 24x3y  To Multiply Monomials To find an equivalent expression for the product of two monomials, multiply the coefficients and then multiply the variables using the product rule for exponents

4. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 4 Bruce Mayer, PE Chabot College Mathematics From §1.6  Exponent Properties 1 as an exponenta 1 = a 0 as an exponenta 0 = 1 Negative exponents The Product Rule The Quotient Rule The Power Rule(a m ) n = a mn Raising a product to a power (ab) n = a n b n Raising a quotient to a power This summary assumes that no denominators are 0 and that 00 00 is not considered. For any integers m and n

5. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Multiply Monomials  Multiply: a) (6x)(7x) b) (5a)(−a) c) (−8x 6 )(3x 4 )  Solution a) (6x)(7x) = (6  7) (x  x) = 42x 2  Solution b) (5a)(−a) = (5a)(−1a) = (5)(−1)(a  a) = −5a 2  Solution c) (−8x 6 )(3x 4 ) = (−8  3) (x 6  x 4 ) = −24x 6 + 4 = −24x 10

6. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 6 Bruce Mayer, PE Chabot College Mathematics (Monomial)(Polynomial)  Recall that a polynomial is a monomial or a sum of monomials. Examples of polynomials: 5w + 8, −3x 2 + x + 4, x, 0, 75y 6  Product of Monomial & Polynomial To multiply a monomial and a polynomial, multiply each term of the polynomial by the monomial.

7. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  (mono)(poly)  Multiply: a) x & x + 7 b) 6x(x 2 − 4x + 5)  Solution a) x(x + 7) = x  x + x  7 = x 2 + 7x b) 6x(x 2 − 4x + 5) = (6x)(x 2 ) − (6x)(4x) + (6x)(5) = 6x 3 − 24x 2 + 30x

8. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  (mono)(poly)  Multiply: 5x 2 (x 3 − 4x 2 + 3x − 5)  Solution: 5x 2 (x 3 − 4x 2 + 3x − 5) = = 5x 5 − 20x 4 + 15x 3 − 25x 2

9. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 9 Bruce Mayer, PE Chabot College Mathematics Product of Two Polynomials  To multiply two polynomials, P and Q, select one of the polynomials, say P. Then multiply each term of P by every term of Q and combine like terms.

10. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  (poly)(poly)  Multiply x + 3 and x + 5  Solution (x + 3)(x + 5) = (x + 3)x + (x + 3)5 = x(x + 3) + 5(x + 3) = x  x + x  3 + 5  x + 5  3 = x 2 + 3x + 5x + 15 = x 2 + 8x + 15

11. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  (poly)(poly)  Multiply 3x − 2 and x − 1  Solution (3x − 2)(x − 1) = (3x − 2)x − (3x − 2)1 = x(3x − 2) – 1(3x − 2) = x  3x − x  2 − 1  3x − 1(−2) = 3x 2 − 2x − 3x + 2 = 3x 2 − 5x + 2

12. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  (poly)(poly)  Multiply: (5x 3 + x 2 + 4x)(x 2 + 3x)  Solution: 5x 3 + x 2 + 4x x 2 + 3x 15x 4 + 3x 3 + 12x 2 5x 5 + x 4 + 4x 3 5x 5 + 16x 4 + 7x 3 + 12x 2

13. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  (poly)(poly)  Multiply: (−3x 2 − 4)(2x 2 − 3x + 1)  Solution 2x 2 − 3x + 1 −3x 2 − 4 −8x 2 + 12x − 4 −6x 4 + 9x 3 − 3x 2 −6x 4 + 9x 3 − 11x 2 + 12x − 4

14. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 14 Bruce Mayer, PE Chabot College Mathematics PolyNomial Mult. Summary  Multiplication of polynomials is an extension of the distributive property. When you multiply two polynomials you distribute each term of one polynomial to each term of the other polynomial.  We can multiply polynomials in a vertical format like we would multiply two numbers (x – 3) (x – 2) x _________ + 6 –2x + 0–3xx2x2 _________ x2 x2 –5x + 6

15. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 15 Bruce Mayer, PE Chabot College Mathematics PolyNomial Mult. By FOIL  FOIL Method  FOIL Example (x – 3)(x – 2) =x 2 – 5x + 6x(x)+ x(–2)+ (–3)(x) + (–3)(–2) =

16. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 16 Bruce Mayer, PE Chabot College Mathematics FOIL Example  Multiply (x + 4)(x 2 + 3)  Solution F O I L (x + 4)(x 2 + 3) = x 3 + 3x + 4x 2 + 12 O I F L = x 3 + 4x 2 + 3x + 12  The terms are rearranged in descending order for the final answer FOIL applies to ANY set of TWO BiNomials, Regardless of the BiNomial Degree

17. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 17 Bruce Mayer, PE Chabot College Mathematics More FOIL Examples  Multiply (5t 3 + 4t)(2t 2 − 1)  Solution: (5t 3 + 4t)(2t 2 − 1) = 10t 5 − 5t 3 + 8t 3 − 4t = 10t 5 + 3t 3 − 4t  Multiply (4 − 3x)(8 − 5x 3 )  Solution: (4 − 3x)(8 − 5x 3 ) = 32 − 20x 3 − 24x + 15x 4 = 32 − 24x − 20x 3 + 15x 4

18. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 18 Bruce Mayer, PE Chabot College Mathematics Special Products  Some pairs of binomials have special products (multiplication results).  When multiplied, these pairs of binomials always follow the same pattern.  By learning to recognize these pairs of binomials, you can use their multiplication patterns to find the product more quickly & easily

19. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 19 Bruce Mayer, PE Chabot College Mathematics Difference of Two Squares  One special pair of binomials is the sum of two numbers times the difference of the same two numbers.  Let’s look at the numbers x and 4. The sum of x and 4 can be written (x + 4). The difference of x and 4 can be written (x − 4). The Product by FOIL: (x + 4)(x – 4) =x 2 – 4x + 4x – 16 = x 2 – 16 ()

20. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 20 Bruce Mayer, PE Chabot College Mathematics Difference of Two Squares  Some More Examples (x + 4)(x – 4) =x2 x2 – 4x + – 16 =x2 x2 – (x + 3)(x – 3) =x2 x2 – 3x + – 9 =x2 x2 – 9 (5 – y)(5 + y) =25 +5y – 5y – y2 y2 =25 – y2y2 } What do all of these have in common?  ALL the Results are Difference of 2-Sqs: Formula → (A + B)(A – B) = A 2 – B 2

21. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 21 Bruce Mayer, PE Chabot College Mathematics General Case F.O.I.L.  Given the product of generic Linear Binomials (ax+b)·(cx+d) then FOILing: Can be Combined IF BiNomials are LINEAR

22. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 22 Bruce Mayer, PE Chabot College Mathematics Geometry of BiNomial Mult  The products of two binomials can be shown in terms of geometry; e.g, (x+7)·(x+5) → (Length)·(Width) 35 5x5x 7x7xx2x2 Width = (x+5) Length = (x+7)  (Length)·(Width) = Sum of the areas of the four internal rectangles

23. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Diff of Sqs  Multiply (x + 8)(x − 8)  Solution: Recognize from Previous Discussion that this formula Applies (A + B)(A − B) = A 2 − B 2  So (x + 8)(x − 8) = x 2 − 8 2 = x 2 − 64

24. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Diff of Sqs  Multiply (6 + 5w)(6 − 5w)  Solution: Again Diff of 2-Sqs Applies → (A + B)(A − B) = A 2 − B 2  In this Case A  6&B  5w  So (6 + 5w) (6 − 5w) = 6 2 − (5w) 2 = 36 − 25w 2

25. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 25 Bruce Mayer, PE Chabot College Mathematics Square of a BiNomial  The square of a binomial is the square of the first term, plus twice the product of the two terms, plus the square of the last term.  (A + B) 2 = A 2 + 2AB + B 2  (A − B) 2 = A 2 − 2AB + B 2 These are called perfect-square trinomials

26. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Sq of BiNomial  Find: (x + 8) 2  Solution: Use (A + B) 2 = A 2 +2  A  B + B 2 (x + 8) 2 = x 2 + 2  x  8 + 8 2 = x 2 + 16x + 64

27. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Sq of BiNomial  Find: (4x − 3x 5 ) 2  Solution: Use (A − B) 2 = A 2 − 2  A  B + B 2  In this Case A  4x&B  3x 5 (4x − 3x 5 ) 2 = (4x) 2 − 2  4x  3x 5 + (3x 5 ) 2 = 16x 2 − 24x 6 + 9x 10

28. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 28 Bruce Mayer, PE Chabot College Mathematics Summary  Binomial Products  Useful Formulas for Several Special Products of Binomials: For any two numbers A and B, (A − B) 2 = A 2 − 2AB + B 2 For two numbers A and B, (A + B) 2 = A 2 + 2AB + B 2 For any two numbers A and B, (A + B)(A − B) = A 2 − B 2.

29. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 29 Bruce Mayer, PE Chabot College Mathematics Multiply Two POLYnomials 1.Is the multiplication the product of a monomial and a polynomial? If so, multiply each term of the polynomial by the monomial. 2.Is the multiplication the product of two binomials? If so: a)Is the product of the sum and difference of the same two terms? If so, use pattern (A + B)(A − B) = A 2 − B 2

30. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 30 Bruce Mayer, PE Chabot College Mathematics Multiply Two POLYnomials 2.Is the multiplication the product of Two binomials? If so: b)Is the product the square of a binomial? If so, use the pattern (A + B) 2 = A 2 + 2AB + B 2, or (A − B) 2 = A 2 − 2AB + B 2 c)c) If neither (a) nor (b) applies, use FOIL 3.Is the multiplication the product of two polynomials other than those above? If so, multiply each term of one by every term of the other (use Vertical form).

31. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 31 Bruce Mayer, PE Chabot College Mathematics Example  Multiply PolyNoms a) (x + 5)(x − 5) b) (w − 7)(w + 4) c) (x + 9)(x + 9) d) 3x 2 (4x 2 + x − 2) e) (p + 2)(p 2 + 3p – 2)  SOLUTION a) (x + 5)(x − 5) = x 2 − 25 b)(w − 7)(w + 4) = w 2 + 4w − 7w − 28 = w 2 − 3w − 28

32. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 32 Bruce Mayer, PE Chabot College Mathematics Example  Multiply PolyNoms  SOLUTION c) (x + 9)(x + 9)= x 2 + 18x + 81 d) 3x 2 (4x 2 + x − 2) = 12x 4 + 3x 3 − 6x 2 e) By columns p 2 + 3p − 2 p + 2 2p 2 + 6p − 4 p 3 + 3p 2 − 2p p 3 + 5p 2 + 4p − 4

33. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 33 Bruce Mayer, PE Chabot College Mathematics Function Notation  From the viewpoint of functions, if f(x) = x 2 + 6x + 9 and g(x) = (x + 3) 2  Then for any given input x, the outputs f(x) and g(x) above are identical.  We say that f and g represent the same function

34. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 34 Bruce Mayer, PE Chabot College Mathematics Example  f(a + h) − f(a)  For functions f described by second degree polynomials, find and simplify notation like f(a + h) and f(a + h) − f(a)  Given f(x) = x 2 + 3x + 2, find and simplify f(a+h) and simplify f(a+h) − f(a)  SOLUTION f (a + h) = (a + h) 2 + 3(a + h) + 2 = a 2 + 2ah + h 2 + 3a + 3h + 2

35. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  f(a + h) − f(a)  Given f(x) = x 2 + 3x + 2, find and simplify f(a+h) and simplify f(a+h) − f(a)  SOLUTION f (a + h) − f (a) = [(a + h) 2 + 3( a + h) + 2] − [a 2 + 3a + 2] = a 2 + 2ah + h 2 + 3a + 3h + 2 − a 2 − 3a − 2 = 2ah + h 2 + 3h

36. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 36 Bruce Mayer, PE Chabot College Mathematics Multiply PolyNomials as Fcns  Recall from the discussion of the Algebra of Functions The product of two functions, f·g, is found by (f·g)(x) = [f(x)]·[g(x)]  This can (obviously) be applied to PolyNomial Functions

37. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  Fcn Multiplication  Given PolyNomial Functions  Then Find: (f·g)(x) and (f·g)(−3)  SOLUTION (f · g)(x) = f(x) · g(x) (f · g)( − 3)

38. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 38 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §5.2 Exercise Set 30, 54, 82, 98b, 116, 118  Perfect Square Trinomial By Geometry

39. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 39 Bruce Mayer, PE Chabot College Mathematics All Done for Today Remember FOIL By BIG NOSE Diagram

40. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 40 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

41. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 41 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

42. BMayer@ChabotCollege.edu MTH55_Lec-21_sec_5-2_Mult_PolyNoms.ppt 42 Bruce Mayer, PE Chabot College Mathematics