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Today’s Objective (1)

Today’s Objective (1). 1.) To be able to line up the like terms of 2 polynomials. 2.) To be able to add and subtract polynomials. Terms to know. Monomial - a number, variable, or product of either with only exponents of 0 or positive integers.

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Today’s Objective (1)

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  1. Today’s Objective (1) • 1.) To be able to line up the like terms of 2 polynomials. • 2.) To be able to add and subtract polynomials.

  2. Terms to know • Monomial- a number, variable, or product of either with only exponents of 0 or positive integers. y, -x, ab, (1/3)x, x2, 8, xy2, (abc2)3

  3. Special Note • Monomial- No monomial has a variable as an exponent, nor does it have a variable in the denominator of a fraction. • 3/y, xa

  4. Terms to know • Polynomial- is the sum or difference of monomials. • Any Monomial is also a polynomial • a-b, x, -2x2 +xy-3, • 1/8x - xy2, r + 9, 6 Examples

  5. Adding Polynomials • Add 5x + 7 and 8 - 2x (5x + 7) (-2x + 8) + = 3x + 15 or

  6. AddingPolynomials • Add 5x + 7 and 8 - 2x line up the 5x + 7 -2x + 8 like terms + 3x + 15

  7. Subtracting Polynomials subtract3a + bfrom7a + 5b (7a + 5b) (3a + b) = - 7a + 5b -3a - b = 7a -3a + 5b - b 4a + 4b or

  8. Subtracting Polynomials • Subtract3a + bfrom7a + 5b line up the (7a + 5b) (3a + b) like terms - 4a + 4b

  9. Adding Polynomials • Add c2 + 5c + 4 and 3c - 7 c2 + 5c + 4 3c - 7 + = c2 + 5c + 3c + 4 - 7 c2 + 8c - 3 or

  10. Adding Polynomials • Add c2 + 5c + 4 and 3c - 7 line up the c2 + 5c + 4 3c - 7 like terms + c2 + 8c - 3

  11. Today’s Objective (2) • To understand the similarities and differences of: • Monomials (1) • Binomials (2) • Trinomials (3) • Polynomials (Many)

  12. Monomials • Have one term such as: • 6, 7a, 5x2, -4m3n2 Why is a monomial? -4m3n2

  13. Binomials • Have two terms such as • 5x + 3, 6y2 - 2, • a - b, 2x2y - 3xy2 The terms are separated by one operation sign (+ or -) Notice:

  14. Trinomials • Have three terms such as: • 3x2 + 5x - 6 • -3m + m3 -2 The terms are separated by two operation signs (+ or -) Notice:

  15. Be ready to answer the following questions: • 1.) What separates the terms of a polynomial? • 2.) How many signs separate the termsof a trinomial? + or - operation signs 2

  16. Which of these are monomials? • 1.) x2 y2, x2 /y2, 1/7, ax2 + bx + c, 1/x + y Why aren't the others Monomials?

  17. Which of these are Polynomials? • 1.) x2 + y2, x3, x2 - 1/3, ax2 + bx + c, 1/x + y Why isn't 1/x + y a polynomial?

  18. Classifying Polynomials • Polynomials are Classified by degree. • The Degree is determined by the exponents of the terms. For example:

  19. The degree of a Monomial Monomial Degree • Is the sum of the exponents of the variables of the monomial. x3 3 x3 y2 5 3x3 y2 5 32x3 y2 5

  20. The degree of a Monomial Monomial Degree • Is the sum of the exponents of the variables of the monomial. 9 0 x1 xy2

  21. The degree of a Polynomial • Is the highest degree of any of its terms after the poly has been simplified. Polynomial Degree 2 3x2 + 5x + 7

  22. The degree of a Polynomial Polynomial Degree 2 3x2 + 5x + 7 3 3x2 -9xyz +y+z 1 x + y + 7 2x2 +7x -3y-2x2 2

  23. ascending going up the stairs

  24. descending going down the stairs

  25. Descending order of Polynomials • From the highest degree to the lowest degree of the terms. • 3x2 + 5x + 7 • 3x3 + 5x2 - 2x + 7 1 0 2 0 1 3 2

  26. Ascending order of Polynomials • From the lowest degree to the highest degree of the terms. • 7 + 5x + 3x2 • 7 - 2x + 5x2 +3x3 0 1 2 3 0 1 2

  27. Today’s Objective (3) Learn Basic Laws of Exponents Whenever we have variables which contain exponents and have equal bases, we can do certain mathematical operations to them. Those operations are called the “Laws of Exponents” bx is read ”b to the x power” b = base x = exponent

  28. If a is a real number and n is a positive integer,

  29. Basic Laws of Exponents

  30. Other Properties of Exponents Any single number or variable is always to the first power

  31. Basic Examples

  32. More Examples

  33. More Examples

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