Operating Polynomials Section 5.3 12/5/2018 6:00 PM 5.3 - Operating Polynomials

Add/Subtract Polynomials Keep It Simple, Students Combine like terms when adding or subtracting polynomials Create a table/chart when simplifying polynomials (for the visual learners) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 1 Add (x2 – 5x + 6) + (3x2 + 4x – 9) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Your Turn Add (3x3 – y + 2x – 5) + (x + y + 5) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 2 Subtract (x2 – 5x + 6) – (3x2 + 4x – 9) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Your Turn Subtract (7x2 + 4x + 7) – (–8x + 2) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 3 Multiply 5x(2x3 + 6) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Your Turn Multiply –5x3(2x2 – 9x + 2) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Multiplying Factors FOIL Method: Multiply the first term in the first bracket and first term in the second bracket (FIRST) Multiply the first term in the first bracket and second term in the second bracket (OUTER) Multiply the second term in the first bracket and first term in the second bracket (INNER) Multiply the second term in the first bracket and second term in the second bracket (LAST) Simplify the expression 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 4 (using FOIL) Multiply (x + 2)(x + 3) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 4 (using Box) Multiply (x + 2)(x + 3) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 5 Multiply (4x + 3)(3x – 5) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Your Turn Multiply (x4 + 2)(3x2 – 1) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Review Add (x – 3) + (–x2 + 2x + 4) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

Example 6 (using Distribution) Multiply (x – 3)(–x2 + 2x + 4) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 6 (using Box) Multiply (x – 3)(–x2 + 2x + 4) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 7 Multiply (x + 2)(3x2 – 4x + 1) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Your Turn Multiply (2x3 – 3x + 4)(x2 + 1) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 8 Multiply (x – 5)(x + 1)(x + 3) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 8 Multiply (x – 5)(x + 1)(x + 3) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Your Turn Multiply (2x + 1)(3x – 2)(4x – 3) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Pascal’s Triangle Named after Blaise Pascal, a famous French Mathematician and Philosopher Taken from the Fibonnacci’s sequence Viewed through the process of factoring variable binomials 12/5/2018 6:00 PM 5.3 - Operating Polynomials

Steps in Binomial Expansion Plug the equation using Binomial Theorem, using Pascal’s Triangle to determine the coefficients Follow the countdown method for the parenthesis The coefficients follow Pascal's Triangle The first given number or variable increase left to right The second number or variable decreases left to right Simplify 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Pascal’s Triangle (x + y)0 = 1 Row 0 (x + y)0 1 (x + y)1 = x + y Row 1 (x + y)1 1 1 (x + y)2 = x2 + 2xy + y2 1 2 1 Row 2 (x + y)2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 1 3 3 1 Row 3 (x + y)3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 1 4 6 4 1 Row 4 (x + y)4 (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 1 5 5 1 10 10 Row 5 (x + y)5 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 9 Multiply (x + y)4 1 2 3 4 6 5 10 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 9 1 1 1 Multiply (x + y)4 1 2 1 1 3 3 1 1 4 6 4 1 1 5 5 1 10 10 4 3 2 2 3 1 4 x + 4 x y 6 x y 4 1 + + x y + y 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 10 Multiply (x – 5)3 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 11 Multiply (2x – 1)3 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Your Turn Multiply (3x – 4)3 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Example 12 Mr. Silva manages a manufacturing plant. From 1990 through 2005 the number of units produced (in thousands) can be modeled by N(x) = 2x2 + 2x + 3. The average cost per unit (in dollars) can be modeled by C(x) = –4x2 –.1x + 3. Write a polynomial T(x) that can be used to model the total costs. Total cost is the product of the number of units and the cost per unit. T(x) = N(x)  C(x) 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Your Turn The number of items is modeled by 0.3x2 + 0.1x + 2, and the cost per item is modeled by g(x) = –0.1x2 – 0.3x + 5. Write a polynomial c(x) that can be used to model the total cost. Total cost is the product of the number of units and the cost per unit. y = –0.03x4 – 0.1x3 + 1.27x2 – 0.1x + 10 12/5/2018 6:00 PM 5.3 - Operating Polynomials

5.3 - Operating Polynomials Assignment Worksheet 12/5/2018 6:00 PM 5.3 - Operating Polynomials