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polynomials

polynomials

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polynomials

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  1. polynomials

  2. DEFinitions • Monomials have one term • Examples would be? • 18 or x or 18x or 18x2y • Can be a constant, a variable(s) or a product of a constant and variable(s) • Polynomials have more than one term • Examples would be? • 18x – 1 or 18x2y – 18x – 1 • Binomials have 2 terms • Trinomials have 3 terms

  3. What are SIMilar terms?What are like terms? • Terms that are exactly alike except for the coefficient • EXAMPLE? 5n and n 18x2y and x2y BUT NOT 5n and n2 • To simplify a polynomial you combine like (similar) terms • EXAMPLE -3x2 + 5x – 2x + x2 - 4 • SIMPLIFIED -2x2 + 3x - 4

  4. Practice! • SIMPLIFY: 4m2 + 5mn2 – m2 + 3mn2 • ANSWER: 3m2 + 8mn2

  5. practice • Write the sum of the area of the rectangles as a polynomial in its simplest form 2x 5 5x 5x Area of a rectangle = lw Area A + Area B + Area C + Area D 2x 2x 2x 5

  6. practice • ADD: 2a2 + ab + 2b and 4a2 – 3ab + 9 Hint: you can set it up so that you can see the like terms. • ANSWER 2a2 + ab + 2b +4a2 – 3ab + 9 6a2 – 2ab + 2b + 9

  7. practice • SUBTRACT: 2x2 – y2 from 5x2 + 7xy + 2y2 Hint: you can set it up so that you can see the like terms. • ANSWER 5x2 + 7xy + 2y2 -(2x2 - y2) 3x2 + 7xy + 3y2

  8. degree • The degree of a monomial – the sum of the degrees of its variables. • EXAMPLE 5xy2z Degree of 4 • The degree of a polynomial – the greatest of the degrees of its terms after it has been simplified. • EXAMPLE -3x2 + 5x – 2x + x2 - 4 Simplified -2x2+ 3x - 4 degree of 2

  9. Multiplying polynomials by monomials

  10. Multiplying polynomials by monomials What is x+3 multiplied by x? *****This is a polynomial (x+3) multiplied by a monomial x***** x(x+3)=x2+3x

  11. Let’s Prove it using geometry x+3 x 3 • x x x Area = lw Area = (x)(x) Area = x2 Area = lw Area = (3)(x) Area = 3x Area = lw Area = x(x+3) Area of rectangle = x(x+3) Area of rectangle = x2 + 3x x(x+3)=x2 + 3x

  12. practice • SIMPLIFY y(y-2) • ANSWER y2-2y

  13. practice • Simplify -3b(2b2 + 4b -1) • ANSWER -6b3 – 12b2 + 3b

  14. practice • SIMPLIFY a[2a – 3(1 – a)] • ANSWER a[2a – 3 + 3a] a[5a-3] 5a2 - 3a

  15. Multiplying polynomials by polynomials

  16. Multiplying polynomials What is x+3 multiplied by x+2? *****This is a polynomial (x+3) multiplied by another polynomial (x+2)***** (x+3)(x+2)=x2+2x+3x+6 x2+5x+6

  17. Let’s Prove it using geometry x+3 x 3 Area = lw Area = (3)(x) Area = 3x • x • x X+2 Area = lw Area = (3)(2) Area = 6 2 2 Area = lw Area = (x)(x) Area = x2 Area = lw Area = (2)(x) Area = 2x Area = lw Area = (x+2)(x+3) Area of rectangle = x2 + 3x + 2x + 6 Area of rectangle = x2 + 5x + 6 Area of rectangle = x2 + 2x + 3x + 6Area of rectangle = x2 + 5x + 6 (x+2)(x+3)=x2 + 5x + 6

  18. Remember “foil” for binomials! first, outside, inside, last (x+1)(x+2) X2 + 2x + 1x + 2 So…x2 + 3x + 2

  19. Practice – use foil • MULTIPLY (y+4)(y-1) • ANSWER y2-y + 4y - 4 simplify y2 + 3y - 4

  20. Practice – use foil • MULTIPLY (2x+6)(3x-5) • ANSWER 6x2- 10x +18x - 30 simplify 6x2 + 8x – 30

  21. Practice – with trinomials! • MULTIPLY (2-s)(3 + 5s -4s2) • ANSWER 6 +10s – 8s2 -3s -5s2 + 4s3 Simplify 4s3 – 13s2 + 7s + 6

  22. Practice – with trinomials! • MULTIPLY (x + 2y)(3x2 - y2+xy) • ANSWER 3x3 -xy2 + x2y + 6x2y – 2y3 + 2xy2 SIMPLIFY: 3x3+ 7x2y + xy2 – 2y3