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11. 1 Multiplying a Monomial by a Polynomial NOTES MULTIPLYING A MONOMIAL BY A POLYNOMIAL monomial- no + or - signs ex/ 5x2y3 b inomial- ONLY 1 + or - sign ex/ 5x3+3y trinomial- ONLY 2 + and/or - signs ex/ 7x2+6x+4 polynomial- 3 or more +/- signs ex/ 5x2+y2+3x+2y-7 STEPS: To distribute term outside ( ) to every term inside ( ) do the following: 1. Multiply the # outside ( ) by each # of each term inside ( ) 2. Add the exponents of the variables in every term outside ( ) to the same variables in every term inside ( ) 3. Write your answer in standard form (put terms in order from highest to lowest C B A A 2a(a2-3a+1) C -3x3(2x4-x2-8) B 1 4 x3(8x2+12x-4) A 1 2 x2y(4x+2xy-8y) A BELLWORK 1. 3x (4x2 - 3x + 2) 2. -b4 (4b3 + b2 - 2b) 3. 12a2b (2ab2 + 1a - 3b) 11. 2 Multiplication of Binomials (baby bear - c levels) NOTES MULTIPLICATION OF BINOMIALS BOX MODEL: (x+3)(2x-4) F.O.I.L: First Outer Inner Last (2x-4)(x+3) (x+3)(2x-4) Examples: C C C Examples: (2a-5)(a+4) C (x-8)(3x-4) C (2m2-6m)(m+2) C BELLWORK 11. 3 F.O.I.L. (mama bear - b levels) SQUARING FOIL (Special Cases) SQUARING BINOMIALS: - When given a binomial squared, you must re-write to FOIL ex/ (3x-2)2 ex/ ( 4x+3)2 B B Plus/Minus: - FOIL just like always :-) BUT - notice a patternBthat the middle term will cancel out ex/ (x-2)(x+2) ex/ (3a2+j)(3a2-j) B B BELLWORK 11. 4 Multiplying Polynomials (papa bear - a levels) NOTES & EX #1 Multiplying Polynomials Each term from the first binomial should multiply to each term of the trinomial. Ex1: (2x2-1)(4x2-3x+2) A Box Method: 2 A 2 Ex2: (4x +5)(2x -5x+1) Ex3: 3x4+(2x2-6)2 In HW BELLWORK 11. 5 Factoring the Greatest Common Factor Finding the GCF FINDING THE GREATEST COMMON FACTOR Finding the GCF: 1. 12, 6 2. 9x2, 15x 3. 30x4, 6x6, 18x2 4. x5y3, x8y2 5. 10a3b4c, 5ab3, 20a2b Factoring a Polynomial: C 1. 7x-14 C 2. 16x-8 B 3. 5x3+10x2-15x B 4. 22x3+x5+14x7 5. 28x2y+14xy2- 7xy A BELLWORK 11. 6 Solving by Factoring Out the Greatest Common Factor NOTES & EX 1 Solving By Factoring Out the Greatest Common Factor STEPS: 1. Set the equation equal to 0 2. Find the factors & GCF 3. Set each factor equal to 0 and solve Ex1: 3x-6=0 C C B A Ex2: 2x+10=0 C Ex3: 3x2-6x=0 B Ex4: -4x2=12x A BELLWORK 11.8 7 11. Factor and Solving (Quadratics) Factor and Solve Trinomials Special Trinomials NOTES & EX 1 Factoring & Solving Trinomials (Quadratics) STEPS: 1. Factor out the GCF if there is one 2. Factor the trinomial using factors of "c" that have a "b" 3. Write as 2 binomials & FOIL to check answer 4. Set each binomial equal to 0 and solve. Ex 1: x2+9x+18 sum of B,C B,C A Ex 2: x2+25x+24 B,C Ex 3: x2-11x+30 B,C Ex 4: 5x2-30x+45 A BELLWORK 11. 8 Factor and Solve Special Trinomials B Levels FOLDED PAPER INSERT (C Levels) STAPLE C-Level Paper Examples Factoring & Solve Special Trinomials FACTOR & SOLVE: (B Level) 1. 25x -4=0 2. 16x2-81=0 3. 4x2-16=0 5. 9x2-81=0 2 A Level The area of a rectangle is represented by x2-169 L a) Write the expression for the length & width of the rectangle. L =W W= b) If x=16 ft, what is L & W? L= W= BELLWORK 11.9 Factor and Solve by Grouping Ex 1 FOLDED PAPER INSERT (Steps) STAPLE Steps: (For polynomials with 4 terms, do not combine the like terms) To Factor: 1.Look at the terms of the polynomial and see if you can factor out a GCF. 2.Group the first two and last two terms of the polynomial together. 3.Factor out the GCF separately for the terms grouped together. (At this point you should have 2 matching binomials.) 4.The remaining binomials should be the same. If not, see if you can factor out a negative of one of the groups to create common binomials. 5.The two GCFs come together to form a binomial and the common binomial is the other factor of the polynomial. To Solve: 1. Set each binomial equal to zero. 2. Solve each to find the solutions Factor & Solve by Grouping 1. 4x2+12x-x-3=0 2. 2x2+9x+5x+20=0 3. 9x2+36x-6x-24=0 BELLWORK 11.10 Quadratics with Leading Coefficients >1 Notes & Example Quadratics with Leading Coefficients >1 Steps: 1. Multiply "a" times "c" 2. Find the factors of that product that add to "b" 3. Re-write trinomial as polynomial & follow steps from 11.9 1. 4x2+11x-3=0 B,C A 2. 3x2+10x-8=0 B,C 3. 4x2+26x+40=0 A BACK COVER SIDE 4. 18x2-20x-3=9x+12 BELLWORK