7 3, 36. 5, 2, and π are all constants.

Transcription

1 Chapter Section 1 Lesson Monomials Introduction This lesson introduces monomials, exponents, and associated terminology. Definitions Like any other subject, algebra has its own vocabulary sets of words that are specific to the subject. To understand algebra, you must learn its vocabulary. A variable is a letter or symbol used to represent a quantity that is unknown or can change. The letters x and y are the symbols most commonly used as variables but any letter can be used. Variables are also sometimes referred to as "unknowns." Common nouns can serve this purpose in the English language. For example, the word cat represents different cats; the variable x represents different numbers. A constant is a quantity that does not change in value. For example,, 8, 7, 6. 5,, and π are all constants. A monomial is a constant, a variable, or the product of constants and variables. A monomial never involves addition, subtraction, radicals of variables, or variables in a denominator For example,, 9xy, u v w, and a b are all monomials. The following are not monomials: 9x + y, x, 5 x, 1 x, and 5. x Based on the descriptions above, try to answer the following questions yourself before looking at the answers. Question: Is 7 b a monomial? Yes, because 7 is a constant, b is a variable, and 7 b is their product. Question: Is 7 b + x a monomial? No. It is the sum of two monomials. Question: Is xy a monomial? No. Monomials can never have variables under the radical sign. Page 1 of 1

2 Example A 7b x a monomial? Is ( ) In this case, 7b and x are factors to be multiplied: 7b( x) = 1bx. (Remember, when you multiply, the variables "go along for the ride.") The result is the product of the constant 1 and the variables b and x. So it is a monomial. Notes: The monomial 1 bx is written in standard form. This means that the constant comes first and the variables come second, in alphabetical order, when writing the product. In the question above, 7b x is a monomial, but it is not in standard form. ( ) When a monomial is written in standard form, the constant is called the coefficient of the monomial. In the monomial 1 bx, 1 is the coefficient. Try to answer the following questions yourself before looking at the answers. Question: Write 5 m q in standard form and find the coefficient. The standard form is 10 mq, and the coefficient of the monomial is 10. Question: Write n( 5x)( c) in standard form, and find the coefficient. The standard form is 0cnx, and the coefficient is 0. Question: What is the coefficient of xyz? Although no number appears in the monomial, the coefficient is xyz = xyz. 1( ) 1 because Question: Write x ( y) ( z) in standard form, and find the coefficient. Multiply the numbers ( ) ( ) ( 1). The variables go along for the ride: 6xyz. The coefficient is 6. Page of 1

3 Exponential Notation Products of monomials with repeating factors such as x x notation is used. The product of x x is written as x, and read as x to the second power. So, x x = x x = x, indicating that x is used as a factor two times. The repeated factor, x, is called the base. are not written as xx. Instead, a special The number of repeated factors,, is called the exponent. Note: The expression x can also be read as x squared, and the expression to the third power" or x cubed. Examples: x x x x x = x 5 and is read x to the fifth power and is read y to the third power y y y = y Consider the three ways of writing : = = 81 is said to be in exponential form is in expanded form 81 is in standard form x can be read as " x Question: Write 65 in exponential form, as a power of 5. Ask yourself: "Raising 5 to what power equals 65?" 65 = , so there are four factors of 5 : 65 = 5. Degree The degree of a monomial with only one variable is simply the degree of that variable. Examples: The degree of 5x is. 9 The degree of y is 9. The degree of x is 1 since x 1 = x. The degree of a constant is 0. Page of 1

4 The following illustration may help you understand some of the terminology we ve just studied: Example B What is the standard form of x x x = x x = x x x = x The standard form is x. The degree of x, and what is this monomial s degree? x is, since x is used as a factor three times. Example C Write the monomial x 7 x in standard form, and identify the coefficient, variable, and degree. Standard Form Coefficient Variable Degree x 7 x = 1x 1 x Example D Write the monomial y 5 y y in standard form, and identify the coefficient, variable, and degree. Standard Form Coefficient Variable Degree y 5 y y = 0y 0 y Question: Write the monomial 5a a in standard form; state its coefficient and degree. Multiply the coefficients: 5 = 15, which is the coefficient of the monomial in standard form. Now find the number of factors of a : there are + = 5 factors of a. So 5a a = 15a, with coefficient 15 and degree 5. 5 Page of 1

5 Example E Write the monomial x y xy in standard form. 6 It might help to reorganize the expression by putting the numbers and variables next to each other and in alphabetical order. 6 x y xy = 6 x x y y = x y Since x is used as a factor times and y is used as a factor times, x y is the standard form of the given monomial. Example F Write the product of 7x y, xyz, and yz as a monomial in standard form. x, factors of Since 7 = 8, the coefficient of the product is 8. There are factors of 5 y, and 5 factors of z, so: 7x y xyz yz = 8x y z. 5 The monomial 8x y z is in standard form. Example G Write the product of the monomials a c, 6ab, bc, and abc as a monomial in standard form. (Remember that 1 is the coefficient of abc). There are factors of a, factors of b, and 6 factors of c. The coefficient is 6 1 =. 6 a c 6ab bc abc = a b c Example H Write the product of the monomials xy, ( ) = ( ) 10x z, and wxz in standard form. xy 10x z wxz 8 10 w x x x y z = 80wx yz Notes: When a product involves negative monomials, use parentheses to help keep things clear when multiplying. The sign of a monomial in standard form is the sign of its coefficient. z Page 5 of 1

6 Extended Example 1a z Write ( ) x y x y z x in standard form. = ( ) Hint: Put all the variables into alphabetical order, to the right of the coefficient you just found. = x x x yy zz Hint: Find the number of factors of x, y, and z. Then, write the product in standard form. There are 5 factors of x, factors of y, and factors of z. The standard form is 5 x y z. 5 z. That is: ( ) x y x y z xz = x y Extended Example 1b in standard form. Write x y ( xy z ) ( 5x z ) 5 = ( ) ( ) 0 = 0x xx y y z z Hint: Find the number of factors of x, y, and z. Then, write the product in standard form. There are 6 factors of x, factors of y, and 5 factors of z. The standard form is 6 5 0x y z. z. 6 5 x y xy z 5x z = 0x y That is, ( ) ( ) Page 6 of 1

7 Extended Example 1c Write ( ) ( ) ( ) u v w 5uv 7u vw in standard form. 5 7 = 105 ( ) ( ) ( ) = 105u uu v v vww Hint: Find the number of factors of u, v, and There are 8 factors of u, 6 factors of v, and factors of w. 8 6 The standard form is 105u vw. ( ) ( ) ( ) 8 6 That is, u v w 5uv 7u vw = 105u v w. w. Then, write the product in standard form. Extended Example a Write A 5 B AB) ( )( )( in standard form and note the coefficient. Hint: Multiply the coefficients ( )( 5)( 1). 5 1 = 0 ( )( )( ) Hint: Put all the variables into alphabetical order, to the right of the coefficient you just found. = 0A AB B Hint: Find the number of factors of A and B and write the product in standard form. = 0A B The coefficient is 0. Page 7 of 1

8 Extended Example b Write ( A )( 9B)( A B ) in standard form, and note the coefficient. 1 9 = 18 ( ) = 18A A BB Hint: Find the number of factors of each variable and write the product in standard form. 5 5 = 18A B The coefficient is 18. Extended Example c Write ( c d )( cd)( 10c d ) in standard form, and note the coefficient = 0 ( ) ( ) ( ) = 0c cc d dd Hint: Find the number of factors of each variable and write the product in standard form. 6 7 = 0c d The coefficient is 0. Page 8 of 1

9 Extended Example a Write ( 5u v 5 w ) ( 0u v w ) in standard form and note the coefficient. 5 0 = 100 ( )( ) = 100u uvvww 5 Hint: Find the number of factors of u, v, and = 100u vw The coefficient is 100. w, and write the product in standard form. Extended Example b r s t 7r s t in standard form, and note the coefficient. Write ( ) ( ) 7 = 1 ( ) ( ) = 1r r s s t t Hint: Find the number of factors of each variable and write the product in standard form = 1r s t The coefficient is 1. Page 9 of 1

10 Extended Example c 8r s t 6 9r 5 s t in standard form, and note the coefficient. Write ( ) ( ) 9 8 = 7 ( )( ) 5 6 = 7r r s s t t Hint: Find the number of factors of each variable and write the product in standard form = 7r s t The coefficient is 7. Extended Example a 6 Write ( x )( x )( 7x)( x ) in standard form. Note the coefficient and the degree. 7 = 8 ( )( )( )( ) Hint: Put the variables to the right of the coefficient. 6 = 8x x xx Hint: Find the number of factors of x and write the product in standard form = 8 x = 8 x The coefficient is 8, and the degree is 1. Page 10 of 1

11 Extended Example b Write ( x 7 )( 9x 5 )( x )( 5 x ) in standard form. Note the coefficient and the degree. 9 5 = 70 ( ) ( ) ( ) ( ) Hint: Put the variables to the right of the coefficient. 7 5 = 70 x x x x Hint: Find the number of factors of x and write the product in standard form = 70 x 16 = 70 x The coefficient is 70, and the degree is 16. Extended Example c Write ( y ) ( 6y 6 ) ( y)( y ) in standard form. Note the coefficient and the degree. 6 = 1 ( )( )( )( ) Hint: Put the variables to the right of the coefficient. 6 = 1 y y yy Hint: Find the number of factors of y and write the product in standard form = 1y 1 = 1 y The coefficient is 1, and the degree is 1. Page 11 of 1

12 Extended Example 5a 11 a b ( 10a b ) Simplify ( ). Write the answer in standard form. 10 = 0 ( ) Hint: Put the variables to the right of the coefficient. 0 a a b b 11 = Hint: Find the number of factors of 6 1 = 0a b a and b and write the product in standard form. Extended Example 5b Simplify ( 11ab)( 9ab). Write the answer in standard form = 99 ( ) Hint: Put the variables to the right of the coefficient = 99a abb Hint: Find the number of factors of each variable and write the product in standard form. = 99a = 99a b 11 1 b Page 1 of 1

13 Extended Example 5c Simplify ( 100 x 1 y 10 )( 17 x 0 y 1 ). Write the answer in standard form = 1700 ( ) ( ) Hint: Put the variables to the right of the coefficient = 1700 x y Hint: Find the number of factors of each variable and write the product in standard form. = 1700 x y End of Lesson Page 1 of 1