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SLIDE SHOW INSTRUCTIONS This presentation is completely under your control. This lesson will show only one step at a time, to see the next step you must press a key. (Actual names written on a key are in green) •TO STOP THE SLIDE SHOW: press ‘escape’ (Esc, top left of keyboard) •TO MOVE FORWARD: press the “spacebar” or Enter (PageDn, , , also work) •TO MOVE BACKWARD: press the key (PageUp, or also work) Polynomials: Like Terms A word about “math words” Before we go through the procedure for combining like terms we are going to define a few “math words” Often, the instructions for homework problems are given, but since the student does not know what the “math words” mean, the instructions are misunderstood or ignored. The words we need to know are defined on the next two slides. Take a moment to write the words and definitions down on a piece of paper so you can refer to them as we go along. Monomials, Polynomials & Terms Definitions: A Constant is a single number (with no variables) Examples of Constants: 2, -5, 1/2, 4.083 Notice that a constant can be an integer, a fraction, a decimal, a square root, etc. A Variable is a letter that represents an unknown number Examples of Variables: x, x2, y, a5 Variables can be alone, in groups, or raised to powers A Term is a group of numbers and variables which are being multiplied. Examples of Terms: 2s, -5xy , 3, x3y2z, t3 Note: When a number is part of a term, it is called a Coefficient Example: 4x2 has a coefficient of 4 Terms are also called Monomials Monomials, Polynomials & Terms Definitions: A Monomial is a single term (mono means ‘one’) Examples of monomials: 2, -5, 3x, 5x2y, xyz A monomial can be made up of numbers, variables, or both. A Polynomial is made up of two or more terms The Terms of a polynomial are separated by plus or minus signs Examples of polynomials: 2x + 2, x2 – 5y, 3x + 2y - 9 TWO TERMSTWO TERMS THREE TERMS Some polynomials have special names: Binomials have exactly two terms (Bi means two) Trinomials have exactly three terms (Tri means three) Examples of Monomials and Polynomials -5x3y x2 + 2 is a monomial is a binomial (two terms) -6x2 - 4x + 2 is a trinomial (three terms) 2x3 + 4x2 - x + 5 is a 4-term polynomial -x3 + 5x2 - 7xy + 5x - 1 is a 5-term polynomial Notice that the CONSTANT is always at the end, and the VARIABLES are listed from highest power to lowest and are also in alphabetical order. This is called DESCENDING ORDER Descending Order & Degree -x3 + 5x2 - 7xy + 5x - 1 is a 5-term polynomial Notice that -7xy is before 5x in the last example. This is because the degree of the term -7xy is higher than the degree of the 5x term. The DEGREE OF A TERM is found by adding up the powers of the variables in a term. So, the degree of -7x1y1 is 2 (2nd degree) and the degree of +5x1 is 1 (1st degree) Combining Like Terms LIKE TERMS are groups of terms which have exactly the same variables (including their powers). The coefficients (numbers in front) can be the same or different: LIKE TERMS can be combined by adding or subtracting the coefficients. The variables remain the same. In the polynomial: -3x + 4xy + x2 - 2x, the only set of like terms are -3x and -2x (both have x’s) So combining like terms, we get: -3x - 2x = -5x So when we combine like terms in the polynomial: -3x + 4xy + x2 - 2x We get the answer: -5x +4xy + x2 putting the terms in descending order, we get: x2 + 4xy - 5x Polynomial Addition is combining Like Terms LIKE TERMS are groups of terms which have exactly the same variables (including their powers). The coefficients (numbers in front) can be the same or different: In the two polynomials: x2 - 3x + 4 and -4x2 - 2y - 5 There are two sets of like terms: x2 and -4x2 and +4, -5 1x2- 4x2 = -3x2 and +4 - 5 = -1 (notice that only the coefficient changes, not the variable or the power) LIKE TERMS can be combined by adding or subtracting the coefficients. The variables remain the same. So when we combine like terms in the polynomials: x2 - 3x + 4 and -4x2 - 2y - 5 We get the answer: -3x2 - 3x - 2y - 1 Polynomial Addition is combining Like Terms In the polynomial: x2y2 - 3x2y + 4xy2 - 2, there are no like terms. Even though three terms have xy’s in them, the powers are different. Remember the variables and the powers must match in order to be called like terms. So since we can’t combine like terms in the polynomial: x2y2 - 3x2y + 4xy2 - 2 The original is the answer: x2y2 - 3x2y + 4xy2 - 2 Practice Problems: (Hit enter to see the answers) Combine like terms 1) -6x2 + 2x - 7x2- x 5) 2) 5xy + 2x - 3xy - 2 6) -3y2 + 2y2 - y2 + 1 3) ab + 2a2b 7) 2xy - 5x - 3xy + 6 - 7x 4) 3x3y - x3y + 2x2y 8) - x - x - x + 3x 5 + 2x - 3 + 4x + 2 Answers: 1) -13x2 + x 2) 2x + 2xy - 2 3) ab + 2a2b (or 2a2b + ab) 4) 2x3y + 2x2y 5) 4 + 6x (or 6x + 4) 6) -2y2 + 1 7) -xy - 12x + 6 8) 0 (3x - 3x = 0) End of Tutorial Go to www.greenebox.com for more great math tutorials for your home computer Questions? send e-mail to: [email protected]