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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. 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 A Constant is a single number (with no variables) Examples of Constants: 2, -5, 1/2, 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, x 2, y, a 5 Definitions: A Term is a group of numbers and variables which are being multiplied. Examples of Terms: 2s, -5xy, 3, x 3 y 2 z, t 3 Variables can be alone, in groups, or raised to powers Monomials Terms are also called Monomials Note: When a number is part of a term, it is called a Coefficient Example: 4x 2 has a coefficient of 4
A Monomial is a single term (mono means one) Examples of monomials: 2, -5, 3x, 5x 2 y, 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, x 2 – 5y, 3x + 2y - 9 Some polynomials have special names: Binomials have exactly two terms (Bi means two) Trinomials have exactly three terms (Tri means three) Definitions: TWO TERMS THREE TERMS Monomials, Polynomials & Terms
Examples of Monomials and Polynomials monomial -5x 3 y is a monomial binomial x is a binomial (two terms) 4-term polynomial 2x 3 + 4x 2 - x + 5 is a 4-term polynomial trinomial -6x 2 - 4x + 2 is a trinomial (three terms) 5-term polynomial -x 3 + 5x 2 - 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 - 7xy + 5x 5-term polynomial -x 3 + 5x 2 - 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. So, the degree of -7x 1 y 1 is 2 (2nd degree) and the degree of +5x 1 is 1 (1st degree) The DEGREE OF A TERM is found by adding up the powers of the variables adding up the powers of the variables in a term.
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 polynomial: -3x + 4xy + x 2 - 2x, the only set of like terms are -3x and -2x (both have xs) So combining like terms, we get: -3x - 2x = -5x adding or subtracting the coefficients LIKE TERMS can be combined by adding or subtracting the coefficients. The variables remain the same. So when we combine like terms in the polynomial: -3x + 4xy + x 2 - 2x We get the answer: -5x +4xy + x 2 putting the terms in descending order, we get: x 2 + 4xy - 5x
Like Terms 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: x 2 - 3x + 4 and -4x 2 - 2y - 5 There are two sets of like terms: x 2 and -4x 2 and +4, -5 1x 2 - 4x 2 = -3x 2 and = -1 (notice that only the coefficient changes, not the variable or the power) adding or subtracting the coefficients 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: x 2 - 3x + 4 and -4x 2 - 2y - 5 We get the answer: -3x 2 - 3x - 2y - 1
Like Terms Polynomial Addition is combining Like Terms In the polynomial: x 2 y 2 - 3x 2 y + 4xy 2 - 2, no like terms there are no like terms. Even though three terms have xys in them, the powers are different. Remember the variables and the powers must match in order to be called like terms. So since we cant combine like terms in the polynomial: x 2 y 2 - 3x 2 y + 4xy The original is the answer: x 2 y 2 - 3x 2 y + 4xy 2 - 2
Practice Problems: (Hit enter to see the answers) Combine like terms 1) -6x 2 + 2x - 7x 2 - x 5) 5 + 2x x + 2 2) 5xy + 2x - 3xy - 2 6) -3y 2 + 2y 2 - y ) ab + 2a 2 b 7) 2xy - 5x - 3xy x 4) 3x 3 y - x 3 y + 2x 2 y 8) - x - x - x + 3x Answers: 1) -13x 2 + x 2) 2x + 2xy - 2 3) ab + 2a 2 b (or 2a 2 b + ab) 4) 2x 3 y + 2x 2 y 5) 4 + 6x (or 6x + 4) 6) -2y ) -xy - 12x + 6 8) 0 (3x - 3x = 0)
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