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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. §5.2 Multiply PolyNomials. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 5.1. Review §. Any QUESTIONS About §5.1 → PolyNomial Functions Any QUESTIONS About HomeWork §5.1 → HW-15. Multiply Monomials.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Chabot Mathematics §5.2 MultiplyPolyNomials Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 5.1 Review § • Any QUESTIONS About • §5.1 → PolyNomial Functions • Any QUESTIONS About HomeWork • §5.1 → HW-15

  3. Multiply Monomials • Recall Monomial is a term that is a product of constants and/or variables • Examples of monomials: 8, w, 24x3y • To Multiply MonomialsTo 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. From §1.6 Exponent Properties This summary assumes that no denominators are 0 and that 00 is not considered. For any integers m and n

  5. Example  Multiply Monomials • Multiply: a) (6x)(7x) b) (5a)(−a) c) (−8x6)(3x4) • Solution a) (6x)(7x) = (6  7) (x  x) = 42x2 • Solution b) (5a)(−a) = (5a)(−1a) = (5)(−1)(a  a) = −5a2 • Solution c) (−8x6)(3x4) = (−8  3) (x6  x4) = −24x6 + 4 = −24x10

  6. (Monomial)•(Polynomial) • Recall that a polynomial is a monomial or a sum of monomials. • Examples of polynomials:5w + 8, −3x2 + x + 4, x, 0, 75y6 • Product of Monomial & Polynomial • To multiply a monomial and a polynomial, multiply each term of the polynomial by the monomial.

  7. Example  (mono)•(poly) • Multiply: a) x & x + 7 b) 6x(x2− 4x + 5) • Solutiona) x(x + 7) = x x + x  7 = x2 + 7x b) 6x(x2− 4x + 5) = (6x)(x2) −(6x)(4x) + (6x)(5) = 6x3− 24x2 + 30x

  8. Example  (mono)•(poly) • Multiply: 5x2(x3 − 4x2 + 3x− 5) • Solution:5x2(x3 − 4x2 + 3x− 5) = = 5x5− 20x4 + 15x3− 25x2

  9. 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. 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 = x2 + 3x + 5x + 15 = x2 + 8x + 15

  11. 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) = 3x2− 2x− 3x + 2 = 3x2− 5x + 2

  12. Example  (poly)•(poly) • Multiply: (5x3 + x2 + 4x)(x2 + 3x) • Solution: 5x3 + x2 + 4x x2 + 3x 15x4 + 3x3 + 12x2 5x5 + x4 + 4x3 5x5 + 16x4 + 7x3 + 12x2

  13. Example  (poly)•(poly) • Multiply: (−3x2− 4)(2x2− 3x + 1) • Solution 2x2− 3x + 1 −3x2− 4 −8x2+ 12x−4 −6x4 + 9x3− 3x2 −6x4 + 9x3−11x2 + 12x−4

  14. 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 PolyNomial Mult. Summary (x – 3) (x – 2) x _________ –2x + 6 _________ x2 –3x + 0 x2 –5x + 6

  15. FOIL Method PolyNomial Mult. By FOIL • FOIL Example (x – 3)(x – 2) = x(x) + x(–2) + (–3)(x) + (–3)(–2) = x2 – 5x + 6

  16. L F I O FOIL Example FOIL applies to ANY set of TWO BiNomials, Regardless of the BiNomial Degree • Multiply (x + 4)(x2+ 3) • Solution F O I L (x + 4)(x2+ 3) = x3 + 3x + 4x2 + 12 = x3 + 4x2 + 3x + 12 • The terms are rearranged in descending order for the final answer

  17. More FOIL Examples • Multiply (5t3 + 4t)(2t2− 1) • Solution:(5t3 + 4t)(2t2− 1) = 10t5− 5t3 + 8t3− 4t = 10t5 + 3t3− 4t • Multiply (4 − 3x)(8 − 5x3) • Solution: (4 − 3x)(8 − 5x3) = 32 − 20x3− 24x + 15x4 = 32 − 24x− 20x3 + 15x4

  18. 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. 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: x2 – 16 ( ) (x + 4)(x – 4) = x2 – 4x + 4x – 16 =

  20. Difference of Two Squares • Some More Examples } (x + 4)(x – 4) = x2 – 4x + 4x – 16 = x2 – 16 What do all of these have in common? (x + 3)(x – 3) = x2 – 3x + 3x – 9 = x2 – 9 (5 – y)(5 + y) = 25 +5y – 5y – y2 = 25 – y2 • ALL the Results are Difference of 2-Sqs:Formula → (A + B)(A –B) = A2–B2

  21. 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. x2 7x 5x Geometry of BiNomial Mult • The products oftwo binomials can be shown in terms of geometry; e.g,(x+7)·(x+5) →(Length)·(Width) Length= (x+7) 35 Width= (x+5) • (Length)·(Width) = Sum of the areas of the four internal rectangles

  23. Example  Diff of Sqs • Multiply (x + 8)(x− 8) • Solution: Recognize from Previous Discussion that this formula Applies(A + B)(A −B) = A2−B2 • So (x + 8)(x− 8) = x2− 82 = x2− 64

  24. Example  Diff of Sqs • Multiply (6 + 5w)(6 − 5w) • Solution: Again Diff of 2-Sqs Applies → (A + B)(A −B) = A2−B2 • In this Case • A 6 & B  5w • So (6 + 5w) (6 − 5w) = 62− (5w)2 = 36 − 25w2

  25. 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 = A2 + 2AB + B2 • (A−B)2 = A2− 2AB + B2 These are called perfect-square trinomials

  26. Example  Sq of BiNomial • Find: (x + 8)2 • Solution: Use (A + B)2 = A2+2AB + B2 (x + 8)2 = x2 + 2x8 + 82 = x2 + 16x + 64

  27. Example  Sq of BiNomial • Find: (4x− 3x5)2 • Solution: Use (A − B)2 = A2 −2AB + B2 • In this Case • A 4x & B  3x5 (4x− 3x5)2 = (4x)2− 2  4x  3x5 + (3x5)2 = 16x2− 24x6 + 9x10

  28. Summary  Binomial Products • Useful Formulas for Several Special Products of Binomials: For any two numbers A and B, (A + B)(A −B) = A2 − B2. For two numbers A and B, (A + B)2 = A2 + 2AB + B2 For any two numbers A and B, (A − B)2 = A2− 2AB + B2

  29. Multiply Two POLYnomials • Is the multiplication the product of a monomial and a polynomial? If so, multiply each term of the polynomial by the monomial. • Is the multiplication the product of two binomials? If so: • Is the product of the sum and difference of the same two terms? If so, use pattern(A + B)(A−B) = A2−B2

  30. Multiply Two POLYnomials • Is the multiplication the product of Two binomials? If so: • Is the product the square of a binomial? If so, use the pattern (A + B)2 = A2 + 2AB + B2, or (A−B)2 = A2− 2AB + B2 • c) If neither (a) nor (b) applies, use FOIL • 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. Example  Multiply PolyNoms a) (x + 5)(x− 5) b) (w− 7)(w + 4) c) (x + 9)(x + 9) d) 3x2(4x2 + x− 2) e) (p + 2)(p2 + 3p– 2) • SOLUTION • (x + 5)(x− 5) = x2− 25 • (w− 7)(w + 4) = w2 + 4w− 7w− 28 = w2− 3w− 28

  32. Example  Multiply PolyNoms • SOLUTION • (x + 9)(x + 9) = x2 + 18x + 81 • 3x2(4x2 + x− 2) = 12x4 + 3x3− 6x2 e) By columns p2 + 3p− 2 p + 2 2p2 + 6p− 4 p3 + 3p2− 2p p3 + 5p2 + 4p− 4

  33. Function Notation • From the viewpoint of functions, if f(x) = x2 + 6x + 9 and g(x) = (x + 3)2 • Then for any given input x, the outputsf(x) and g(x) above are identical. • We say that f and g represent the same function

  34. 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) = x2 + 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 = a2+ 2ah + h2 + 3a + 3h + 2

  35. Example  f(a + h) − f(a) • Given f(x) = x2 + 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] − [a2 + 3a + 2] =a 2+ 2ah + h 2 + 3a + 3h + 2 −a2− 3a− 2 = 2ah + h2+ 3h

  36. 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. Example  Fcn Multiplication • Given PolyNomial Functions • Then Find: (f·g)(x) and (f·g)(−3) • SOLUTION (f·g)(−3) (f·g)(x) = f(x) ·g(x)

  38. WhiteBoard Work • Problems From §5.2 Exercise Set • 30, 54, 82, 98b, 116, 118 • PerfectSquareTrinomialByGeometry

  39. All Done for Today Remember FOIL By BIG NOSEDiagram

  40. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

  41. Graph y = |x| • Make T-table

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