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Lesson 4 MA 15200 A Monomial or Term is a number or the product of a number and one or more variables with whole-number exponents. The number factor is called a Coefficient of the term. The Degree of the monomial (or term) is the sum of the exponents of it variables. A Constant has degree 0. Ex 1: For each term, state the coefficient and the degree of the term. a) 14 x 3 y b) c) d) a5 2 r 2h 5.1 A monomial or a sum of monomials is called a Polynomial. A polynomial with 2 terms is called a Binomial, and a polynomial with 3 terms is called a Trinomial. The Degree of the Polynomial is the degree of the term of the polynomial of the highest degree. Ex 2: For each polynomial, state the number of terms and the degree. a) 4 x3 5 x 9 x 4 3x 2 b) 12 x 2 y 7 x 2 y 2 8 xy 2 The leading coefficient of a polynomial is the coefficient of the term with the highest degree. That term is also called the leading term. Ex 3: State the degree of each polynomial. Identify any binomials or trinomials. a) 4 x3 3x 2 y 2 3xy 4 y 2 b) 3 4 ab 2c a 2b 2c 4 3 Ex 4: State the leading term and the leading coefficient of this polynomial. 4 xy 3x 2 y 7 x3 y 8 x 1 Monomials or Terms that have the same variable factors are called like terms and can be combined by combining the coefficients and keeping the same variable factors. Addition/Subtraction of Polynomials: To add or subtract polynomials: 1. Remove parentheses using the distributive property. 2. Combine any like terms. Ex 5: Add/Subtract these polynomials a ) (3a 3b 7 a 2b 4ab) (5a 2b 14a 3b 8ab) b) (4 x3 4 x 2 x 2 6) (2 x 2 9 3 x x 3 ) c) 4(2m3 7 m 2 ) 6(8m 2 m3 ) 2(2m3 3m 2 ) Multiplication of Polynomials: 1. To multiply two monomials: Use regular 'rules of exponents'. Multiply the coefficients and multiply the variables. 2. To multiply a monomial and a polynomial (binomial or more terms): Use the distributive property. Multiply each term of the polynomial by the monomial. 3. To multiply two binomials: Use FOIL (first, outer, inner, last) 4. To multiply two general polynomials with 2 or more terms: Use a variation of FOIL. Distribute each term of the first polynomial to every term of the second polynomial and combine the results. Ex 6: Multiply and write product in simplified form. 1 2 a) (4a 2b) ab 4 c) ( m 2)(3m 5) e) (4 x 3)( x 2 3 x 4) 4 xy 2 (3x 3 2 x 2 y xy 2 ) b) d) ( 2 x)( 5 2 x) 2 *Note: Only two polynomials (other than monomials) can be multiplied at one time. To find the following product, multiply two of the binomials, then multiply that product by the third binomial. ( x 3)( x 4)(2 x 1) Special Products: 1. ( x y )( x y ) x 2 y 2 2. These types of binomials are called conjugates . ( x y ) 2 x 2 2 xy y 2 (1st) 2 2(1st)(2nd) (2nd) 2 3. ( x y ) 2 x 2 2 xy y 2 (1st) 2 2(1st)(2nd) (2nd) 2 Ex 7: Find each product in simplest form. a ) (4m 3n) 2 b) (6 5w)(6 5w) c) (2 x 3) 2 d) (4 y 5)(4 y 5) e) (2 x )(2 x ) 3 The conjugate of a b is a b and the conjugate of a b is a b . The product of conjugates with any square root terms will always be rational (without radicals). To rationalize a denominator with a binomial, multiply numerator and denominator by the conjugate of the denominator. Ex 1: Rationalize the denominator and simplify where possible. 9 a) 6 3 b) 12 x x 1 c) 4 5 3 3 4