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**Unformatted text preview: **MA100, Fall 2010, PART 1: Precalculus [6] Polynomial and algebraic expressions 1. Polynomials, polynomials degree, leading coefficient, operations Definition A monomial is a mathematical expression of the form a X n , where a R \ { } and n N { } . The number a is called the coefficient of the monomial, and the number n is called its degree . Example: (- 2 + 1) X 3 is a monomial. Its degree is 3 and its coefficient is (- 2 + 1) . Definition Let a , a 1 , a 2 , . . . , a n R with a n 6 = 0 . A polynomial of degree n is a mathematical expression obtained as a sum of monomials: a n X n + a n- 1 X n- 1 + . . . + a 2 X 2 + a 1 X + a . The numbers a , a 1 , . . . a n R are called the coefficients of the polynomial and a n 6 = 0 is called the leading coefficient . Example: The polynomial (- 11) X 5- 23 X 3 + 4 X + 1 is of degree 5. The leading coefficient is (- 11). The coefficient of the term of degree 3 is (- 23) . The coefficient of the term of degree 1 is 4 . Notation Usually, polynomials are denoted P ( X ). Evaluating polynomials Given a polynomial P ( X ) = a n X n + a n- 1 X n- 1 + . . . + a 2 X 2 + a 1 X + a , and a number R , the value of P ( X ) in X = is obtained by substituting X by in P ( X ) and it is denoted by P ( , . Example: The value of polynomial P ( X ) = (- 11) X 5- 23 X 3 + 4 X + 1 in X = (- 1) is P (- 1) = (- 11) (- 1) 5- 23 (- 1) 3 + 4 (- 1) + 1 = (- 11) (- 1)- 23 (- 1) + 4 (- 1) + 1 = 22 + 23- 4 + 1 = 42 We write P (- 1) = 42 . Example: Evaluate P ( X ) = 2 X 4- 7 X 3 + 4 X 2 + X in X = 3 . Solution: P (3) = 2 3 4- 7 3 3 + 4 3 2 + 3 = 2 81- 7 27 + 4 9 + 3 = 162 + 189- 36 + 3 = 318 The answer is P (3) = 318 . 1 Addition of polynomials The addition of two polynomials results in a polynomial whos coefficients are obtained by adding the corresponding coefficients of the monomials of the same degree. Example: (- 11) X 5- 23 X 3 + 4 X + 1 + X 7- 4 3 X 5 + 2 X 4 + 5 X 3 + 7 = (- 11) X 5- 23 X 3 + 4 X + 1 + X 7- 4 3 X 5 + 2 X 4 + 5 X 3 + 7 = X 7 +- 11- 4 3 + 2 X 4 + (- 23 + 5) X 3 + 4 X + (1 + 7) = X 7- 11 + 4 3 + 2 X 4- 18 X 3 + 4 X + 8 . Multiplication of monomials The multiplication of two monomials aX n and bX m results in a monomial as follows: a X n b X m = ab X n + m . Note that the coefficients multiply as numbers, whereas for the variables X we use one of the exponential rules. Example: 2 X 5 (- 7) X 12 = (- 14) X 17 Multiplication of polynomials The multiplication of two polynomials results in a polynomial whos coefficients are obtained by multiplying term by term the monomials forming the two polynomials, and then collecting the corresponding coefficients of terms of the same degree....

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