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Section 5.2 Multiplying Polynomials

Section 5.2 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 5.2 Multiplying Polynomials

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  1. Section 5.2Multiplying 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 5.2

  2. Multiplying Two (or more) Monomials Learn todo theseIN YOURHEAD! • Multiply the numeric coefficients • Add exponents of matched variables • Include any unmatched variables • Examples • (3)(2x) = 6x -4y(-2xy) = 8xy2 -2s(r) = -2rs • 3x(2x)(3x) = 18x3 • -5x3(4x2y) = -20x5y • -2(-y) = 2y • (-2b3)(3a)(a2bc) = -6a3b4c Do the variablesin alpha order 5.2

  3. For You 5.2

  4. Multiplying a Polynomial by a Number • Positive numbers – law of distribution • 5 times 2x2 – 3x + 7 • 5(2x2) – 5(3x) + 5(7) Do this in your head? • 10x2 – 15x + 35 • Negative numbers – be careful! • -3 times 4y3 – 6y2 + y – 2 • -3(4y3)– -3(6y2)+ -3(y) - -3(2) In your head? • -12y3 + 18y2 – 3y + 6 5.2

  5. Multiplying a polynomial by a monomial To multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial. 3x2(6xy + 3y2) = 18x3y + 9x2y2 5x3y2(xy3 – 2x2y) = 5x4y5 – 10x5y3 -2ab2(3bz – 2az + 4z3) = -6ab3z + 4a2b2z – 8ab2z3 5.2

  6. Multiplying a Polynomial by a Polynomial (in general) To multiply a polynomial by a polynomial, we use the distributive property repeatedly. Horizontal Method: (2a+ b)(3a – 2b) = 2a(3a – 2b) + b(3a – 2b) = 6a2 – 4ab + 3ab – 2b2 = 6a2 –ab – 2b2 Vertical Method: 3x2 + 2x – 5 4x + 2 6x2 + 4x – 10 12x3 + 8x2 – 20x____ 12x3 + 14x2 – 16x – 10 5.2

  7. Bigger Multiplications Leave Missing Variable Space Leave Margin Space 5.2

  8. FOIL: Used to Multiply Two Binomials 5.2

  9. Multiplying 3 or more Polynomials • Use same technique as you used for numbers: • Multiply any 2 together and simplify the temporary product • Multiply that temporary producttimes any remaining polynomial and simplify • -2r(r – 2s)(5r – s) • = -2r(5r2 – 11rs + 2s2) • = -10r3 + 22r2s – 4rs2 5.2

  10. The Product of Conjugates (Sum and Difference)(A + B)(A – B) = A2 – B2 • The middle term disappears Alwayswhen the binomials are conjugates (identical, except for middle sign) • 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 5.2

  11. Thought provoker:Are these Conjugates? • (x + 2y)(3xz – 6yz) • = 3x2z – 6xyz + 6xyz – 12y2z • = 3x2z – 12y2z • Why does the middle term disappear? • Because the 2nd binomial conceals a conjugate!Both terms contain a common factor, 3z : • (x + 2y)(3xz – 3∙2yz) • The middle term ONLY disappears when binomial conjugates are involved. 5.2

  12. Squaring a Binomial – Creates a Perfect-Square Trinomial(A + B)(A + B) = A2+2AB+ B2 Square the 1st term Multiply 1st times 2nd, double it, add it Square the 2ndterm, add it 5.2

  13. Squaring a Binomial – Creates a Perfect-Square Trinomial(A – B)(A – B) = A2–2AB+ B2 Square the 1st term Multiply 1st times 2nd, double it, subtract it Square the 2ndterm, add it • Differences:Almost thesame 5.2

  14. Examples - board 5.2

  15. Function Notation • If f(x) = x2 – 4x + 5 find: 5.2

  16. Next … • Section 5.3 Intro to FactoringCommon Factors, Factoring by Grouping 5.2

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