영어로 배우는 중학교 수학 - 다항식의 덧셈과 뺄셈 [ 영문 ] - Addition & Subtraction of Polynomials

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  영어로 중학교 수학 - " 다항식의 덧셈과 뺄셈 " 부분을 배워봅시다.    

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   ADDITION  AND  SUBTRACTION  OF  POLYNOMIALS  

We first used polynomials but did not identify them as polynomials.
Polynomials also occured in the equations of previous chapter.  In this section we will
define polynomials and begin a thorough study of polynomials.

Polynomials

In Chapter 1 we defined a term as an expression‎ containing a number or the product of a number and one or more variables raised to powers.  Some examples of terms are

                                                  4x³,  －x²y³ , 6ab , and  －2.

A polynomial is a single term or a finite sum of terms.  The powers of the variables in a polynomial must be positive integers.  For example,

                                                   4x³ + (－15x²) + x + (－2)   

Is a polynomial.  Because it is simpler to write addition of a negative as subtraction, this polynomial is usually written as

                                                    4x³ – 15x² + x – 2.

The degree of a polynomial in one variable is the highest power of the variable in the polynomial.  So 4x³ – 15x² + x – 2 has degree 3 and 7w – w² has degree 2.

The degree of a term is the power of the variable in the term.  Because the last term has no variable, its degree is 0.

A single number is called a constant and so the last term is the constant term.  The degree of a polynomial consisting of a single number such as 8 is 0.

      The number preceding the variable in each term is called the coefficient of that variable or the coefficient of that term.  In 4x³ – 15x²  + x – 2 the coefficient of x³ is 4, the coefficient of x² is –15, and the coefficient of x is 1 because x = 1  ×  x.

E X A M P L E  1            Identifying coefficients 

Determine the coefficients of x³  and x² in each polynomial:

a)  x³ + 5x² – 6                                              b)  4x⁶ – x³ + x

Solution

a )  Write the polynomial as 1 × x³ + 5x² – 6 to see that the coefficient of x³ is 1 and the coefficient of  x²  is  5.

b)   The x²-term is missing in 4x⁶ – x³ + x.  Because 4x⁶ – x³ + x can be written as  

                                               4x⁶  –  1  ×   x³  +  0  ×  x²  +  x,  

the coefficient of x³ is  – 1  and the coefficient of x²  is 0.    

       For simplicity we generally write polynomials with the exponents decreasing from left to right and the constant term last.  So we write

                           x³ – 4x² + 5x + 1         rather than        －4x² + 1 + 5x + x³.

When a polynomial is written with decreasing exponents, the coefficient of the first term is called the leading coefficient. 

   Certain polynomials are given special names.  A monomial is a polynomial that has one term, a binomial is a polynomial that has two terms, and a trinomial is a polynomial that has three terms.  For example, 3x⁵ is a monomial, 2x – 1 is a binomial, and 4x⁶ – 3x + 2 is a trinomial. 

E X A M P L E  2            Types of polynomials

Identify each polynomial as a monomial, binomial, or trinomial and state its degree. 

a)  5x² － 7x³ + 2                 b)  x⁴³ – x²                    c)  5x                 d) －12  

Solution 
a ) The polynomial   5x^{2 –} 7x³ + 2 is a third-degree trinomial.
b ) The polynomial  x⁴³  –  x²  is a binomial with degree 43.
c ) Because 5x = 5x¹, this polynomial is a monomial with degree 1.
d ) The polynomial  –12  is a monomial with degree 0.

Value of a Polynomial

A polynomial is an algebraic expression‎.  Like other algebraic expressions involving variables, a polynomial has no specific value unless the variables are replaced by numbers.  A polynomial can be eval‎uated with or without the function notation discussed in last Chapter.

 E X A M P L E  3          Eval‎uating  Polynomials

a)  Find the value of  – 3x⁴ – x³ + 20x + 3 when x = 1. 

b)  Find the value of  – 3x⁴ – x³ +20x + 3 when x = – 2.

c)  If P(x) = – 3x⁴ – x³ + 20x + 3, find P (1).

Solution 

a)  Replace x by 1 in the polynomial:

                                –3x⁴ – x³ + 20x + 3 = – 3 (1)⁴ – (1)³ + 20(1) + 3

                                                      =  – 3  – 1 + 20 + 3  =  19

b)  Replace x by -2 in the polynomial:

                              –3x⁴ – x³ + 20x + 3 =  – 3( – 2)⁴ – ( – 2)³ + 20( – 2) + 3

                                                   = – 3(16) – ( – 8) – 40 + 3

                                                   = –48 + 8 – 40 + 3 =  – 77

     So the value of the polynomial is – 77 when x = – 2. 

c)  This is a repeat of part (a) using the function notation from Chapter 2. P (1), read

“P of 1,” is the value of the polynomial P(x) when x is 1.  To find P (1), replace x by

1 in the formula for P(x):

                                          P(x) = – 3x⁴ – x³ + 20x + 3

                                          P(1) = – 3 (1)⁴ – (1)³ + 20(1) + 3  =  19 

  So P (1) = 19.  The value of the polynomial when x = 1 is 19.