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Introduction to Polynomials: Understanding Algebraic Terms

Learn the basics of polynomials, from monomials to trinomials, understanding degree and classification, standard form, operations like addition, subtraction, and multiplication, using methods like FOIL and Box. Practice examples included.

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Introduction to Polynomials: Understanding Algebraic Terms

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  1. Polynomials Algebra I Section 9.1 – 9.2 Review

  2. Polynomials Polynomial: A monomial or a sum of monomials. Monomial: 1 term 4xy2 Binomial: 2 terms 5x - 1 Trinomial: 3 terms 3x2 – 2x + 1 Polynomial: 4 terms or more

  3. Degree of a Polynomial To find the degree of a polynomial (if it is a: monomial), add the exponents on the variables contained in each term. Note: If more than one term has the same degree, arrange the terms by which single base has the highest degree. Examples: 1. 8y4 2. 3a 3. -2xy2z3 4. 7

  4. Degree of a Polynomial Constant = 0 Linear = 1 Quadratic = 2 Cubic = 3

  5. Classifying Polynomials –3x5 + 9x3 + 7 Trinomial Degree 5 5x4y6 Monomial Degree 10 –8xy + 15x Binomial Degree 1 6x6 + 2x3 + 3x2 Trinomial Degree 6

  6. Standard Form of a Polynomial Place the terms of the polynomial in order from the least degree to greatest degree. Example: Put the trinomial below in standard form. 9x3 + 7 – 3x5

  7. Polynomial Standard Form Practice 3x2 – 6x2y2 + 4x3 -x4 + 2xy + 7xy4 -9rx + r5x 15km – 13m3 + 10k3

  8. Adding & Subtracting Polynomials (3x2 – 4x + 8) + (2x – 7x2 – 5) Combine “like-terms” and put solution in standard form.

  9. Practice Adding & Subtracting Polynomials 6n2 – 4 (+) -2n2 + 9 (4x + 5xy + 3y) – (3y + 6x + 8xy)

  10. Methods for Multiplying Polynomials by Polynomials 3 Methods: Distribute FOIL Box

  11. Multiplying Polynomials Using Distribution Multiplying Monomials by Polynomials: Use the distributive property. Remember the Rule of Multiplying Numbers with Exponents (add the exponents) Examples: -2x2(3x2 – 7x + 10)

  12. Practice Multiplying Polynomials Using Distribution 4(3d2 + 5d) 5x3(x2 – 2x + 6) 6x(2x2 – 4x) – (x + 2)

  13. FOIL Method FOIL: only works when multiplying a binomial times a binomial. First – 1st terms in each parenthesis Outer – 1st term in 1st parenthesis times last term in 2nd parenthesis Inner – Last term in 1st parenthesis times 1st term in 2nd parenthesis Last – Last term in 1st parenthesis times the last term in 2nd parenthesis

  14. Using FOIL (3x + 5)(x + 6) First Outer Inner Last __________ + _________ + _________ + __________ FOI L

  15. Practice Using FOIL (x + 5)(x + 6) (5m + 2)(7m + 3) (8b – 3)(-2b +6) (9p + 6)(6p – 2)

  16. Box Method Box Method works with all types of polynomials. The size of the box depends on how many terms are in the polynomials. Example: binomial x trinomial = 2 x 3 box

  17. How to use the Box • Put each factor on the outside of a box. • Multiply to fill the box • Combine like-terms (add) to empty the boxes • Be sure to simplify your answer.

  18. How to Use the Box Method (x + 2)(3x2 + 2x + 1)

  19. Practice Using the Box Method (k + 4)(7k2 + 2k – 9) (2x2 – 9x + 11)(2x + 1)

  20. Polynomial Word Problems The length of a pool is (2x2 + 3x) and the width is (x – 2). Find the area of the pool.

  21. Polynomial Word Problems Kenny wants to build a sidewalk around a rectangular garden that is 3 feet wide. If the width of the garden is 2x and the length is 3x + 1, find the area of the sidewalk. 3x + 1 2x 3’

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