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Adding, Subtracting, and Multiplying Polynomials: Techniques and Examples

Explore the essential techniques for adding, subtracting, and multiplying polynomials, including vertical and horizontal formats. Learn how to combine like terms efficiently and practice with examples such as adding and subtracting polynomials, as well as multiplying binomials and trinomial products. Discover key formulas, including the sum and difference of squares, the square of a binomial, and the cube of a binomial, to simplify your calculations. This resource is ideal for students looking to strengthen their polynomial manipulation skills.

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Adding, Subtracting, and Multiplying Polynomials: Techniques and Examples

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  1. 6.3 Adding, Subtracting, & Multiplying Polynomials p. 338

  2. To + or - , + or – the coeff. of like terms!Vertical format : • Add 3x3+2x2-x-7 and x3-10x2+8. • 3x3 + 2x2 – x – 7 + x3 – 10x2 + 8 Line up like terms • 4x3 – 8x2 – x + 1

  3. Horizontal format : Combine like terms • (8x3 – 3x2 – 2x + 9) – (2x3 + 6x2 – x + 1)= • (8x3 – 2x3)+(-3x2 – 6x2)+(-2x + x) + (9 – 1)= • 6x3 + -9x2 + -x + 8 = • 6x3 – 9x2 – x + 8

  4. Examples: Adding & Subtracting • (9x3 – 2x + 1) + (5x2 + 12x -4) = • 9x3 + 5x2 – 2x + 12x + 1 – 4 = • 9x3 + 5x2 + 10x – 3 • (2x2 + 3x) – (3x2 + x – 4)= • 2x2 + 3x – 3x2 – x + 4 = • 2x2 - 3x2 + 3x – x + 4 = • -x2 + 2x + 4

  5. Multiplying Polynomials: Vertically • (-x2 + 2x + 4)(x – 3)= • -x2 + 2x + 4 * x – 3 3x2 – 6x – 12 -x3 + 2x2 + 4x -x3 + 5x2 – 2x – 12

  6. Multiplying Polynomials : Horizontally • (x – 3)(3x2 – 2x – 4)= • (x – 3)(3x2) • + (x – 3)(-2x) • + (x – 3)(-4) = • (3x3 – 9x2) + (-2x2 + 6x) + (-4x + 12) = • 3x3 – 9x2 – 2x2 + 6x – 4x +12 = • 3x3 – 11x2 + 2x + 12

  7. Multiplying 3 Binomials : • (x – 1)(x + 4)(x + 3) = • FOIL the first two: • (x2 – x +4x – 4)(x + 3) = • (x2 + 3x – 4)(x + 3) = • Then multiply the trinomial by the binomial • (x2 + 3x – 4)(x) + (x2 + 3x – 4)(3) = • (x3 + 3x2 – 4x) + (3x2 + 9x – 12) = • x3 + 6x2 + 5x - 12

  8. Some binomial products appear so much we need to recognize the patterns! • Sum & Difference (S&D): • (a + b)(a – b) = a2 – b2 • Example: (x + 3)(x – 3) = x2 – 9 • Square of Binomial: • (a + b)2 = a2 + 2ab + b2 • (a - b)2 = a2 – 2ab + b2

  9. Last Pattern • Cube of a Binomial • (a + b)3 = a3 + 3a2b + 3ab2 + b3 • (a – b)3 = a3 - 3a2b + 3ab2 – b3

  10. Example: • (x + 5)3 = a = x and b = 5 x3 + 3(x)2(5) + 3(x)(5)2 + (5)3 = x3 + 15x2 + 75x + 125

  11. Assignment

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