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 TypeExampleDegree Monomial-10r 6 s 8 14 Binomial29x x Trinomial9p 7 – 4p 3 + 8p 2 7 Polynomial5a 3 b 7 – 3a 5 b 5 + 4a 2 b 9 – a 10 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. (2y 4 – 3y 2 + y) + (4y 4 + 7y 2 + 6y) (-3m 3 – 8m 2 + 4) – (m 3 + 7m 2 – 3)

 Add or subtract, as indicated. 8m 4 p 5 – 9m 3 p 5 + (11m 4 p m 3 p 5 ) 4(x 2 – 3x + 7) – 5(2x 2 – 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 (3p 2 – 4p + 1)(p 3 + 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) = x 2 – y 2 Squares of Binomials (x + y) 2 = x 2 + 2xy + y 2 (x – y) 2 = x 2 – 2xy + y 2 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. (5m 3 + 3)(5m 3 – 3) (3x – 7y 4 ) 2 (9k – 11r 3 )(9k + 11r 3 )