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Section 5.4 Multiplying Polynomials

Section 5.4 Multiplying Polynomials. Multiplying Two Monomials Multiplying a Polynomial By a number By a monomial By another polynomial The FOIL Method Multiplying 3 or More Polynomials Special Products Simplifying Expressions Applications. Multiplying Two (or more) Monomials.

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Section 5.4 Multiplying Polynomials

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  1. Section 5.4 Multiplying Polynomials • Multiplying Two Monomials • Multiplying a Polynomial • By a number • By a monomial • By another polynomial • The FOIL Method • Multiplying 3 or More Polynomials • Special Products • Simplifying Expressions • Applications

  2. Multiplying Two (or more) Monomials • Multiply the signs and numeric coefficients • Examine the variables in alphabetic order • Multiply matched variables by adding exponents • Include any unmatched variables • Examples • (3)(2x) = 6x • -4y(-2xy) = 8xy2 • -2s(r) = -2rs • 3x(2x)(3x) = 18x3

  3. PracticeMultiplying Two (or more) Monomials • Multiply the signs and numeric coefficients • Examine the variables in alphabetic order • Multiply matched variables by adding exponents • Include any unmatched variables • 5x(-x) = -5x2 • 7y(11xy2) = 77xy3 • -4n(m) = -4mn • 3z(-3z)(-2z3) = 18z5 • -2x4(4x3y) = -8x7y • -y(-5) = 5y • (-2b3)(3a)(a2bc) = -6a3b4c

  4. Multiply: a Number and a Polynomial Multiply each polynomial term by that number: • Positive numbers – law of distribution • 5(2x2 – 3x + 7) • 5(2x2) – 5(3x) + 5(7) • 10x2 – 15x + 35 • Negative numbers – be careful! • -3(4y3 – 6y2 + y – 2) • -3(4y3) – (-3)(6y2) + (-3)(y) – (-3)(2) • -12y3 + 18y2 – 3y + 6 • Number following polynomial – other distributive law • (2x3 – 3x2 – 5)(7) • 7(2x3) – (7)(3x2) – (7)(5) • 14x3 – 21x2 – 35

  5. PracticeMultiply: a Number and a Polynomial Multiply each polynomial term by that number: • 5 (-x + 2) = -5x + 10 • 7(11x – 2y2) = 77x – 14y2 • -4(m + 2n) = -4m – 8n • 3(-3z + 5z2 – 2z3) = -9z + 15z2 – 6z3 • -2(4x3 – x2 – x – 3) = -8x3 + 2x2 + 2x + 6 • (x – 9)(3) = 3x – 27 • (-2b3 – 3b + 5)(-5) = 10b3 +15b – 25

  6. Multiply: a Monomial and a Polynomial Multiply each polynomial term by that monomial: • Positive numbers – law of distribution • 3x2(6xy + 3y2) • 3x2(6xy) + 3x2(3y2) • 18x3y+ 9x2y2 • Negative coefficients – be careful! • -2ab2(3bz – 2az + 4z3) • -2ab2(3bz) – (-2ab2)(2az) + (-2ab2)(4z3) • -6ab2z + 4a2b2z – 8ab2z3 • Number following polynomial – other distributive law • (2x3 – 3x2 – 5)(3x) • 3x(2x3) – (3x)(3x2) – (3x)(5) • 6x4 – 9x3 – 15x

  7. Multiply: Two Polynomials Horizontal Method: (use the distributive property repeatedly) (2a+ b)(3a – 2b) = (2a+ b)(3a) –(2a+ b)(2b) = 6a2 + 3ab – (4ab + 2b2) = 6a2 + 3ab – 4ab – 2b2 = 6a2 – ab – 2b2 Vertical Method: 3x2 + 2x – 5 4x + 2Multiply top by rightmost 6x2 + 4x – 10 Then by term to the left 12x3 + 8x2 – 20x____ Lastly, add like terms 12x3 + 14x2 – 16x – 10

  8. Concept: Similar Binomials • Any pair of binomials with matching variable parts • When multiplied, they produce a trinomial or binomial • Examples:

  9. The FOIL MethodUseful for Multiplying Two Similar Binomials (x + 8)(x - 5) = x2 – 5x + 8x – 40 (4t2 + 5)(3t2 - 2) = 12t4 – 8t2 + 15t2– 10 (y – 8)(y2 + 5) = y3 + 5y – 8y2– 40 + 3x + 7t2 – 8y2 + 5y

  10. PracticeUsing the FOIL Method in Your Head If the polynomials are similar, combine the middle terms • (x + 2)(x – 5) = x2 – 3x – 10 • (2y + 3)(4y + 1) = 8y2+ 14y + 3 • (m – 3n)(m – 2n) = m2 – 5mn + 6n2 • (2x + 3y2)(x – 7y2) = 2x2 – 11xy2 – 21y4 • (a + b)(c + d) = ac + ad + bc + bd

  11. Multiplying 3 or more Polynomials • Use same technique as you use for numbers: • Multiply any 2 together and simplify the temporary product • Multiply that temporary product times any remaining polynomial and simplify • -2r(r – 2s)(5r – s) = (use foil on the binomials) • -2r(r2 – 11rs + s2) = (distribute the monomial) • -2r3 – 22r2s – 2rs2

  12. The Product of Conjugates(F + L)(F – L) = F2 – L2 • The middle term disappears ONLY when the binomials are conjugates: identical except for different operations • Multiplying these is easier than using FOIL! • (x + 4)(x – 4) = x2 – 42 = x2 – 16 • (5 + 2w)(5 – 2w) = 25 – 4w2 • (3x2 – 7)(3x2 + 7) = 9x4 – 49 • (-4x – 10)(-4x + 10) = 16x2 – 100 • (6 + 4y)(6 – 4x) = use the foil method= 36 – 24x + 24y – 16xy

  13. Squaring a Binomial Sum(F + L)(F + L) = F2 + 2FL + L2 Square the 1st term Multiply 1st times 2nd, double it, add it Square the 2nd term Try: (2x + 3)2 (2x)2 + 2(6x) + 32 4x2 + 12x + 9 (½x + 5)2 (½x)2 + 2(5x/2) + 52 ¼x2 + 5x + 25

  14. Squaring a Binomial Difference(F – L)(F – L) = F2 – 2FL + L2 Square the 1st term Multiply 1st times 2nd, double it, subtract it Square the 2nd term and add it Try: (3x - 4)2 = (3x)2 – 2(12x) + 42 = 9x2 – 24x + 16 (5a – 2b)2 = (5a)2 – 2(10ab) + (2b)2 = 25a2 – 20ab + 4b2

  15. PracticeBinomial Conjugates and Squares (F + L)(F – L) = F2 – L2(F + L)2 = F2 + 2FL + L2(F–L)2 = F2– 2FL + L2 • (x + 3)(x – 3) = x2 – 9 • (2y – 5)(2y – 5) = 4y2 – 20y + 25 • (m + 3n)2 = m2 + 6mn + 9n2 • (2y – 5)(2y + 5) = 4y2– 25 • (a + b)(a + b) = a2 + 2ab + b2 • (3x – 7y)2 = 9x2 – 42xy + 49y2

  16. What Next? • Present Section 5.5 Factoring by Grouping

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