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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 α ∈ R , the value of P ( X ) in X = α is obtained by substituting X by α in P ( X ) and it is denoted by P ( α, . 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 who’s 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 who’s 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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 Calculus, Algebra, Factoring Polynomials, Polynomials, Mathematical Expression

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