Polynomial Identification and Properties of Exponents

Transcription

1 2 MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS 4a Polynomial Identification and Properties of Exponents Polynomials We begin with the definition of a term. Term. A term is either a single number (called a constant term) or the product of a number and one or more variables. For example, each of the following is a term. 5 3x 2 12y 2 z 3 13a 2 bc 3 Note how the first term is a single number, while the remaining terms are products of a number and one or more variables. For example, 3x 2 is the product of 3, x, and x. Coefficient. When a term is a product of a number and one or more variables, the number is called the coefficient of the term. In the case of a term that is a single number, the number itself is called the coefficient. Thus, for example, the coefficients of the terms are 5, 3, 12, and 13, respectively. 5 3x 2 12y 2 z 3 13a 2 bc 3 Degree. The degree of a term is the sum of the exponents on each variable of the term. A constant term (single number with no variables) has degree zero. Thus, for example, the degrees of the terms 5 3x 2 12y 2 z 3 13a 2 bc 3 are 0, 2, 5, and 6, respectively. In the last example, note that 13a 2 bc 3 is equivalent to 13a 2 b 1 c 3, so adding exponents, we get: Degree of 13a 2 bc 3 =Degreeof13a 2 b 1 c 3 =2+1+3 =6

2 4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF EXPONENTS 3 Monomial. The words monomial and term are equivalent. Thus, are monomials. 5 3x 2 12y 2 z 3 13a 2 bc 3 Binomial. A binomial is a mathematical expression containing exactly two terms. The terms in a mathematical expression are separated by plus or minus signs. For example, each of the mathematical expressions 2x +3y 3a 2 3b 2 xy +7 3x 2 y +5xy 2 is a binomial. Each expression has exactly two terms. Trinomial. A trinomial is a mathematical expression containing exactly three terms. Again, the terms in a mathematical expression are separated by plus or minus signs. For example, each of the mathematical expressions 2x 2 +3x +7 a 2 +2ab + b 2 x 4 2x 2 y 2 +3y 4 is a trinomial. Each expression has exactly three terms. A bicycle has two wheels, a binomial has two terms. A tricycle has three wheels, a trinomial has three terms. But once we get past three terms, the assignment of special names ceases and we use the generic word polynomial, which means many terms. Polynomial. A polynomial is a many-termed mathematical expression. The coefficients of a polynomial are the coefficients of its terms. Each of the previous expressions, 12y 2 z 3 3a 2 3b 2 x 4 2x 2 y 2 +3y 4 though assigned the particular names monomial, binomial, and trinomial, respectively, are also many-termed expressions and can also be called polynomials. However, because the word polynomial means many terms, we can use the word polynomial to describe mathematical expressions with more than three terms, such as: x 4 4x 3 y +6x 2 y 2 4xy 3 + y 4 The coefficients of x 4 4x 3 y +6x 2 y 2 4xy 3 + y 4 are 1, 4, 6, 4, and 1.

3 4 MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS Ascending and Descending Powers When asked to simplify a polynomial expression, we should combine any like terms we find, and when possible, arrange the answer in ascending or descending powers. EXAMPLE 1. Simplify the following polynomial expression, arranging your answer in descending powers of x. Once you ve completed that task, make a second arrangement, ordering your terms in ascending powers of x. 2x 3 +7x 3x 2 +11x +8x x Solution: In order to arrange our answer in descending powers of x, wewant to place the term with the highest power of x first and the term with the lowest power of x last. We use the commutative and associative properties to change the order and regroup, then we combine like terms. 2x 3 +7x 3x 2 +11x +8x x =2x 3 +( 3x 2 +8x 2 )+(7x +11x +15x)+11 =2x 3 +5x 2 +33x +11 Note how the powers of x start at 3, then go down in order. To arrange our final answer in ascending powers of x, we put the lowest power of x first, then the highest power of x last, regrouping and combining like terms. 2x 3 +7x 3x 2 +11x +8x x =11+(7x +11x +15x)+( 3x 2 +8x 2 )+2x 3 =11+33x +5x 2 +2x 3 Note how we start with the constant term, then the powers of x increase in order. When we have a polynomial in a single variable, such as the polynomial in Example 1, arranging the terms in ascending or descending order is fairly straightforward. However, a polynomial in two or more terms is a bit more difficult, and sometimes impossible, to arrange in a decent order. EXAMPLE 2. Simplify the following polynomial expression, then arrange your answer in descending powers of x. x 3 +2xy 2 6x 2 y + y 3 3xy 2 +4x 2 y

4 4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF EXPONENTS 5 Solution: We ll again use the commutative and associative properties to change the order and regroup, putting the terms with the highest powers of x first, then follow with terms containing lowers powers of x in order. x 3 +2xy 2 6x 2 y + y 3 3xy 2 +4x 2 y = x 3 +( 6x 2 y +4x 2 y)+(2xy 2 3xy 2 )+y 3 = x 3 2x 2 y xy 2 + y 3 Note that this is a very natural order, the powers of x decrease while simultaneously the powers of y increase. Not all examples will have nice ordering presented in Example 2, with the powers of one variable descending while the powers of the other variable simultaneously ascends. Sometimes we have to make some very subjective choices on the ordering of terms. EXAMPLE 3. Simplify the following polynomial expression, then arrange your answer in some sort of reasonable order. a 3 b 3 +2a 2 b 3a 2 b 3 +4a 3 b 3 +5a 4 +3a 2 b + b 5 Solution: Let s try to arrange the terms so that the powers of a descend. Again, we use the commutative and associative properties to change the order and regroup. a 3 b 3 +2a 2 b 3a 2 b 3 +4a 3 b 3 +5a 4 +3a 2 b + b 5 =5a 4 +(a 3 b 3 +4a 3 b 3 )+(2a 2 b +3a 2 b) 3a 2 b 3 + b 5 =5a 4 +5a 3 b 3 +5a 2 b 3a 2 b 3 + b 5 Note that in our final arrangement, the powers of a descend, but the powers of b bounce up and down, but at least we have the powers of a descending. That should help us spot if we ve missed a term while simplifying the given problem. The Degree of a Polynomial To find the degree of a polynomial, locate the term of the polynomial having the highest degree. The degree of a polynomial. The degree of a polynomial is the degree of the term having the highest degree.

5 6 MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS Finding the degree of a polynomial of a single variable is pretty easy. EXAMPLE 4. What is the degree of the polynomial x 3 4x x +2x 7? Solution: First, let s arrange the polynomial in descending powers of x. 2x 7 + x 3 4x 2 6x +5 Arranging the polynomial in descending powers of x makes it easier to see that the term of the polynomial with the highest degree is 2x 7.Therefore,the degree of the polynomial is 7. Finding the degree of a polynomial of more than one variable is a little bit harder. EXAMPLE 5. What is the degree of the polynomial x 4 2x 3 y 7 + y 5? Solution: Note that the polynomial is already arranged in descending powers of x, an arrangement that is probably as good as we are going to get. In the following table, we list the degree of each term. Remember, the degree of any term is found by summing the exponents on its variables. Term Degree x 4 4 2x 3 y 7 10 y 5 5 Hence, the term with the highest degree is 2x 3 y 7, making 10 the degree of the polynomial. Laws of Exponents In Chapter 1, section 1, we first introduced the definition of an exponent. For convenience, we repeat that definition. In the exponential expression a n,thenumbera is called the base, while the number n is called the exponent.

6 4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF EXPONENTS 7 Exponents. Let a be any real number and let n be any whole number. If n 0,then: a n = a a a a }{{} n times That is, to calculate a n, write a as a factor n times. In the case where a 0, but n =0,thenwedefine: a 0 =1 For example, raising a number to the fifth power requires that we repeat the number as a factor five times (see Figure??). ( 2) 5 =( 2)( 2)( 2)( 2)( 2) = 32 Raising a number to the fourth power requires that we repeat that number as a factor four times. ( 2) 1 4 ( = 1 )( 1 )( 1 )( 1 ) = 1 16 As a final example, note that 10 0 =1,but0 0 is undefined. Multiplying Like Bases In the expression a n,thenumbera is called the base and the number n is called the exponent. Frequently, we ll be required to multiply two exponential expressions with like bases, suchasx 3 x 4. Recall that the exponent tells us how many times to write each base as a factor, so we can write: x 3 x 4 =(x x x) (x x x x) = x x x x x x x = x 7 Note that we are simply counting the number of times that x occurs as a factor. First, we have three x s, then four x s, for a total of seven x s. However a little thought tells us that it is much quicker to simply add the exponents to reveal the total number of times x occurs as a factor. x 3 x 4 = x 3+4 = x 7 The preceding discussion is an example of the following general law of exponents.

7 8 MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS Multiplying Like Bases. To multiply two exponential expressions with like bases, repeat the base and add the exponents. In symbols: a m a n = a m+n EXAMPLE 6. Simplify each of the following expressions: (a) y 4 y 8 (b) (c) (x + y) 2 (x + y) 7 Solution: In each example we have like bases. Thus, the approach will be the same for each example: repeat the base and add the exponents. (a) y 4 y 8 = y 4+8 (b) =2 3+5 (c) (x + y) 2 (x + y) 7 =(x + y) 2+7 = y 12 =2 8 =(x + y) 9 With a little practice, each of the examples can be simplified mentally. Repeat the base and add the exponents in your head: y 4 y 8 = y 12, =2 8,and (x + y) 2 (x + y) 7 =(x + y) 9.

8 4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF EXPONENTS 9 EXAMPLE 7. Simplify: (a 6 b 4 )(a 3 b 2 ) Solution: We ll use the commutative and associative properties to change the order of operation, then repeat the common bases and add the exponents. (a 6 b 4 )(a 3 b 2 )=a 6 b 4 a 3 b 2 The associative property allows us to regroup in the order we prefer. = a 6 a 3 b 4 b 2 The commutative property allows us to change the order of multiplication. = a 9 b 6 Repeat the common bases and add the exponents. With practice, we realize that if all of the operators are multiplication, then we can multiply in the order we prefer, repeating the common bases and adding the exponents mentally: (a 6 b 4 )(a 3 b 2 )=a 9 b 6. EXAMPLE 8. Simplify: x n+3 x 3 2n Solution: Again, we repeat the base and add the exponents. x n+3 x 3 2n = x (n+3)+(3 2n) Repeat the base, add the exponents. = x 6 n Simplify. Combine like terms. Dividing Like Bases Like multiplication, we will also be frequently asked to divide exponential expressions with like bases, such as x 7 /x 4. Again, the key is to remember that the exponent tells us how many times to write the base as a factor, so wecan write: x 7 x 4 = x x x x x x x x x x x = x x x x x x x x x x x = x 3 Note how we cancel four x s in the numerator for four x s in the denominator. However, in a sense we are subtracting four x s from the numerator, so a

9 10 MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS faster way to proceed is to repeat the base and subtract the exponents, as follows: x 7 x 4 = x7 4 = x 3 The preceding discussion is an example of the second general law of exponents. Dividing Like Bases. To divide two exponential expressions with like bases, repeat the base and subtract the exponents. In symbols: a m a n = am n Note that the subtraction obeys the rule numerator minus denominator. EXAMPLE 9. Simplify each of the following expressions: (a) x12 x 3 (b) (c) (2x +1)8 (2x +1) 3 Solution: In each example we have like bases. Thus, the approach will be the same for each example: repeat the base and subtract the exponents. (a) x12 x 3 = x12 3 (b) = x =57 4 =5 3 (c) (2x +1) 8 =(2x +1)8 3 (2x +1) 3 =(2x +1) 5 With a little practice, each of the examples can be simplified mentally. Repeat the base and subtract the exponents in your head: x 12 /x 3 = x 9,5 7 /5 4 =5 3, and (2x +1) 8 /(2x +1) 3 =(2x +1) 5. EXAMPLE 10. Simplify: 12x 5 y 7 4x 3 y 2 Solution: We first express the fraction as a product of three fractions, the latter two with a common base. In the first line of the following solution,

10 4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF EXPONENTS 11 note that if you multiply numerators and denominators of the three separate fractions, the product equals the original fraction on the left. 12x 5 y 7 4x 3 y 2 = 12 4 x5 x 3 y7 y 2 Break into a product of three fractions. =3x 5 3 y 7 2 Simplify: 12/4=3. Then repeat the common bases and subtract the exponents. =3x 2 y 5 Simplify. x 5n 4 EXAMPLE 11. Simplify: x 3 2n Solution: Again, we repeat the base and subtract the exponents. x 5n 4 = x(5n 4) (3 2n) x3 2n Repeat the base, subtract exponents. = x 5n 4 3+2n Distribute the minus sign. = x 7n 7 Simplify. Combine like terms. Raising a Power to a Power Suppose we have an exponential expression, raised to a second power, such as (x 2 ) 3. The second exponent tells us to write x 2 as a factor three times: (x 2 ) 3 = x 2 x 2 x 2 Write x 2 as a factor three times. = x 6 Repeat the base, add the exponents. Note how we added to get 6. However, a much faster way to add three twos is to multiply: 3 2 = 6. Thus, when raising a power to a second power, repeat the base and multiply the exponents, as follows: (x 2 ) 3 = x 2 3 = x 6 The preceding discussion gives rise to the following third law of exponents. Raising a Power to a Power. When raising a power to a power, repeat the base and multiply the exponents. In symbols: (a m ) n = a mn Note that juxtaposing two variables, as in mn, means m times n.

11 12 MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS EXAMPLE 12. Simplify each of the following expressions: (a) (z 3 ) 5 (b) (7 3 ) 4 (c) [(x y) 3 ] 6 Solution: In each example we are raising a power to a power. Hence, in each case, we repeat the base and multiply the exponents. (a) (z 3 ) 5 = z 3 5 (b) (7 3 ) 4 =7 3 4 (c) [(x y) 3 ] 6 =(x y) 3 6 = z 15 =7 12 =(x y) 18 With a little practice, each of the examples can be simplified mentally. Repeat the base and multiply the exponents in your head: (z 3 ) 5 = z 15,(7 3 ) 4 =7 12, and [(x y) 3 ] 6 =(x y) 18 EXAMPLE 13. Simplify: (x 2n 3 ) 4 Solution: Again, we repeat the base and multiply the exponents. (x 2n 3 ) 4 = x 4(2n 3) Repeat the base, multiply exponents. = x 8n 12 Distribute the 4. Raising a Product to a Power We frequently have need to raise a product to a power, such as (xy) 3. Again, remember the exponent is telling us to write xy as a factor three times, so: (xy) 3 =(xy)(xy)(xy) Write xy as a factor three times. = xyxyxy The associative property allows us to group as we please. = xxxyyy The commutative property allows us to change the order as we please. = x 3 y 3 Invoke the exponent definition: xxx = x 3 and yyy = y 3. However, it is much simpler to note that when you raise a product to a power, you raise each factor to that power. In symbols: (xy) 3 = x 3 y 3 The preceding discussion leads us to a fourth law of exponents.

12 4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF EXPONENTS 13 Raising a Product to a Power. To raise a product to a power, raise each factor to that power. In symbols: (ab) n = a n b n EXAMPLE 14. Simplify each of the following expressions: (a) (yz) 5 (b) ( 2x) 3 (c) ( 3y) 2 Solution: In each example we are raising a product to a power. Hence, in each case, we raise each factor to that power. (a) (yz) 5 = y 5 z 5 (b) ( 2x) 3 =( 2) 3 x 3 (c) ( 3y) 2 =( 3) 2 y 2 = 8x 3 =9y 2 With a little practice, each of the examples can be simplified mentally. Raise each factor to the indicated power in your head: (yz) 5 = y 5 z 5,( 2x) 3 = 8x 3, and ( 3y) 2 =9y 2 When raising a product of three factors to a power, it is easy to show that we should raise each factor to the indicated power. For example, (abc) 3 = a 3 b 3 c 3. In general, this is true regardless of the number of factors. When raising a product to a power, raise each of the factors to the indicated power. EXAMPLE 15. Simplify: (2a 3 b 2 ) 3 Solution: Raise each factor to the third power, then simplify. (2a 3 b 2 ) 3 =2 3 (a 3 ) 3 (b 2 ) 3 Raise each factor to the third power. =8a 9 b 6 Simplify: 2 3 = 8. In the remaining factors, raising a power to a power requires that we multiply the exponents. EXAMPLE 16. Simplify: ( 2x 2 y) 2 ( 3x 3 y)

13 14 MODULE 4. INTEGER EXPONENTS AND POLYNOMIALS Solution: In the first grouped product, raise each factor to the second power. ( 2x 2 y) 2 ( 3x 3 y)=(( 2) 2 (x 2 ) 2 y 2 )( 3x 3 y) =(4x 4 y 2 )( 3x 3 y) Raise each factor in the first grouped product to the second power. Simplify: ( 2) 2 =4and (x 2 ) 2 = x 4. The associative and commutative property allows us to multiply all six factors in the order that we please. Hence, we ll multiply 4 and 3, then x 4 and x 3, and y 2 and y, in that order. In this case, we repeat the base and add the exponents. = 12x 7 y 3 Simplify: (4)( 3) = 12. Also, x 4 x 3 = x 7 and y 2 y = y 3. Raising a Quotient to a Power Raising a quotient to a power is similar to raising a product to a power. For example, raising (x/y) 3 requires that we write x/y as a factor three times. ( ) 3 x = x y y x y x y = x x x y y y = x3 y 3 However, it is much simpler to realize that when you raise a quotient to a power, you raise both numerator and denominator to that power. In symbols: ( ) 3 x = x3 y y 3 This leads to the fifth and final law of exponents. Raising a Quotient to a Power. To raise a quotient to a power, raise both numerator and denominator to that power. In symbols: ( a b ) n = a n b n

14 4A. POLYNOMIAL IDENTIFICATION AND PROPERTIES OF EXPONENTS 15 ou Try It! EXAMPLE 17. Simplify each of the following expressions: ( ) 2 2 ( x ) ( 3 (a) (b) (c) y Solution: In each example we are raising a quotient to a power. Hence, in each case, we raise both numerator and denominator to that power. ) 4 (a) ( ) 2 2 = = 4 9 (b) ( x 3 ) 3 = x = x3 27 (c) ( 2 ) 4 = 24 y y 4 = 16 y 4 Note that in example (c), raising a negative fraction to an even power produces a postive result. With a little practice, each of the examples can be simplified mentally. Raise numerator and denominator to the indicated power in your head: (2/3) 2 =4/9, (x/3) 3 = x 3 /27, and ( 2/y) 4 =16/y 4 EXAMPLE 18. Simplify: ( 2x 2 y 3 ) 2 Solution: Raise both numerator and denominator to the second power, then simplify. ( ) 2x 2 2 = (2x2 ) 2 y 3 (y 3 ) 2 Raise numerator and denominator to the second power. In the numerator, we need to raise each factor of the product to the second power. Then we need to remind ourselves that when we raise a power to a power, we multiply the exponents. = 22 (x 2 ) 2 (y 3 ) 2 Raise each factor in the numerator to the second power = 4x4 y 6 Simplify: 2 2 =4,(x 2 ) 2 = x 4, and (y 3 ) 2 = y 6.