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7.6 Polynomials and Factoring Part 1: Polynomials Basic Terminology • A term, or monomial, is a number, a variable, or a product of numbers and variables. • A polynomial is a term or a finite sum/difference of terms, with only nonnegative integer exponents on the variables. – A polynomial CANNOT have a variable in a denominator. – A polynomial with exactly two terms is a binomial; one with exactly three terms is a trinomial. • The greatest exponent in a polynomial in one variable is the degree of the polynomial. – The degree of a polynomial in more than one variable is equal to the greatest degree of any term appearing in the polynomial. Examples of Various Polynomials Type Monomial Binomial Trinomial Polynomial Example -10r6s8 29x11 + 8x15 9p7 – 4p3 + 8p2 5a3b7 – 3a5b5 + 4a2b9 – a10 Degree 14 15 7 11 Addition and Subtraction • Like terms are terms that have the exact same variable factors. • Polynomials are added by adding coefficients of like terms. • Polynomials are subtracted by subtracting coefficients of like terms. • Polynomials in one variable are often written with their terms in descending powers; so the term of the greatest degree is first, and so on. Adding and Subtracting Polynomials • Add or subtract, as indicated. (2y4 – 3y2 + y) + (4y4 + 7y2 + 6y) (-3m3 – 8m2 + 4) – (m3 + 7m2 – 3) Add or subtract, as indicated. 8m4p5 – 9m3p5 + (11m4p5 + 15m3p5) 4(x2 – 3x + 7) – 5(2x2 – 8x – 4) Multiplication • Polynomials can be multiplied “horizontally” using the Distributive Property. • They can also by multiplied “vertically”. • Both methods work – it’s your choice! Multiplying Polynomials Vertically • Multiply (3p2 – 4p + 1)(p3 + 2p – 8). Using the FOIL Method (Multiplying Horizontally) • The FOIL method is a convenient way to find the product of two binomials. F – first O – outer I – inner L – last • To multiply other polynomials, use the Distributive Property in a similar way. Using the FOIL Method • Find the product. (6m + 1)(4m – 3) Find the product using the method of your choice. (2x + 7)(2x – 7) Special Products • The last example demonstrated a “special product” known as the difference of two squares. Product of the Sum and Difference of Two Terms (x + y)(x – y) = x2 – y2 Squares of Binomials (x + y)2 = x2 + 2xy + y2 (x – y)2 = x2 – 2xy + y2 These are just “short cuts” – If you forget, just multiply it out! Finding Special Products • Find each product. (3p + 11)(3p – 11) (2m + 5)2 Find each product. (5m3 + 3)(5m3 – 3) (3x – 7y4)2 (9k – 11r3)(9k + 11r3)