Adding and Subtracting Polynomials Polynomials: Expressions that may contain numbers, variables, and operations 5a + 8 - 9n + 7a - 6y2 Terms: Parts of a polynomial, separated by addition or subtraction operations 5a + 8 - 9n + 7a - 6y2 Coefficient: The numeric part of a term 5a + 8 - 9n + 7a - 6y2 Constant Term: A term consisting of only a number 5a + 8 - 9n + 7a - 6y2 Like Terms: Terms that share identical variable parts 5a + 8 - 9n + 7a - 6y2 Monomial: A polynomial consisting of a single term Binomial: A simplified polynomial with 2 terms Trinomial: A simplified polynomial with 3 terms
Adding and Subtracting Polynomials Model: Imagine that you are responsible for counting the leftover apples, bananas, and pears after sit-down lunch. You could use the variables; a, b, and p to help you keep track. Simplify: 3a + 4b + p + 2a + b + 6b + 5p + a + 7p = 3a + 2a + a + 4b + b + 6b + p + 5p + 7p = 6a + 11b + 13p Like terms may be combined by adding their coefficients.
Adding and Subtracting Polynomials Now imagine that you and some friends are on a picnic. You brought a nice supply of fruit, but then got hungry . . . You started the day with 9 apples, 7 bananas, and 6 pears, but when it was time for a snack, you ate 5 apples, 1 banana, and 3 pears. Write an expression and simplify it to determine what is left after your snack. Simplify: 9a + 7b + 6p – (5a + b + 3p) = 9a + 7b + 6p – 5a - b - 3p = 9a – 5a + 7b - b + 6p - 3p = 4a + 6b + 3p You have 4 apples, 6 bananas, and 3 pears left over.
Adding and Subtracting Polynomials Variable Terms: Terms that include one or more variables Variable terms may include variables of different degrees. All variables must be of the same degree to be like terms. Remember how x, x2, and x3 may represent line segments, squares, and cubes. They are very different. Simplify: 5 + 4x3 + x2 – (2x3 - x2 – 7x) = 5 + 4x3 + x2– 2x3 + x2+ 7x = 4x3 - 2x3 + x2 + x2 + 7x + 5 = 2x3 + 2x2 + 7x + 5