Lesson #1: Properties of Exponents 1) Multiplying Powers with the Same Base - When multiplying powers that have the same base, add the exponents and keep the base the same. - For example: 3 2 x 3 3 = ( 5) 2 x ( 5) 1 = 3n 4 x 5n 3 3 2+3 = ( 5) 2+1 = (3n x 5n) 4+3 = 3 5 = ( 5) 3 = 15n 4+3 = 243 125 15n 7 2) Dividing Powers with the Same Base - When dividing powers that have the same base, subtract the exponents and keep the base the same. - For example: 4 6 4 3 = 2 9 2 4 = 4 6-3 = 2 9-4 = 4 3 = 2 5 = 64 32 3) Raising Powers, Products, and Quotients to an Exponent - When raising a power to an exponent, multiply the exponents and keep the base the same. - For example: (3 3 ) 2 = (4 2 ) 3 = 3 3x2 = 4 2x3 = 3 6 = 4 6 = 729 4096 - When raising a power to a product, you can multiply the product and then to the exponent or rewrite the product with the same exponent. - For example: (2 x 4) 3 = (2 x 4) 3 = 8 3 = 2 3 x 4 3 = 512 8 x 64 = 512 - When raising a power to a quotient, rewrite the quotient with the same exponent. - For example: (3 4) 3 = (5 3) 2 = 3 3 4 3 = 5 2 3 2 = 27 64 = 25 9 = 0.42 2.78
Assignment #1: Properties of Exponents Simplify each of the following. 1) 3m 2 3m 3 2) m 4 3m -3 3) 5r 3 3r 2 4) 2n 4 5n 3 5) 3k 4 6k 6) 3x 3 y 3 3x 1 y 3 7) 3y 2 4x 8) 3v 3 vu 2 9) 3a 3 b 2 4a 4 b 3 10) x 2 y 4 x 3 y 2 11) (x 2 ) 3 12) (4x 2 ) 4 13) (3r 1 ) 4 14) (3a 3 ) 2 15) (2k 4 ) 4 16) (5xy) 1 17) (4b 4 ) 1 18) (x 2 y 1 ) 2 19) (3x 4 y 3 ) 1 20) (4m) 2 21) 6r 12 3r 3 22) 9x 16 3x 4 23) 2n 4 2n 3 24) 14m 4 7m 4 25) 2m 4 2m 3 26) x 4 y 4 z 3 x 2 y 3 z 4 27) 3xy 2 z 3 3x 28) h 3 g 3 k 4 gk 29) m 4 n 3 p 4 m 2 n 2 p 3
Lesson #2: Combining Expressions Follow the Properties of Exponents discussed in Lesson #1. Here are a few examples of the type of questions you will see in Assignment #2. When a variable has no exponent attached to it, it is actually considered an exponent of one. For example 5n = 5n 1. 14x 2 28x + 35 = 2x 2 4x + 5 7 3x 2 4x 3 + 3x 2 = 4x 3 + 6x 2 (3xy 3 z 2 )(5xyz 3 ) = 15x 2 y 4 z 5 3x 2 3x + x 2 x + 7 = 4x 2 4x + 7 8x 3 2x 3 = 5x 3
Assignment #2: Combining Expressions Solve the following expressions. 1) 7x + 5x 2) 4x 2 3x 3 3) x 5 x 3 4) 6x 3 + x 3 5) 3x 2 4x 3 + 3x 2 6) (3xy 3 z 2 )(5xyz 3 ) 7) 3x 2 3x + x 2 x + 7 8) (x 3 ) 2 9) x 2 + 3x 2 10) 5x 2 2x 3 3x 2 11) (4x 4 yz 2 ) 2 12) ( 3xy 3 ) 3 13) (xy 2 ) 3 (2x 3 y 4 ) 2 14) (3xy)(4x 3 y) 15) x 2 x 3 16) 6x 3 2x 3 17) 5x 3 + 2x 2 3x 3 18) 7x + 3x 2 19) ( 5x 5 ) 2 20) 4xy 2 + 7xy + 3x 2 y 21) 8x + 4y 4 22) x 4x 23) (5xy 3 ) 2 (3y 2 ) 3 24) 7x 4x 25) 6x 2 9x + 18 3 26) ( 2x 3 y) 2 ( 1x 3 y 4 ) 3 27) (4x 5 y 4 )(3xy 3 )
Lesson #3: Adding and Subtracting Polynomials To add polynomials combine all the like (the same) terms. For example: (3x 5) + (5x + 6) 3x and 5x have the same variable, so they are the same 3x + 5x 5 + 6 Grouping like terms 8x + 1 (3n 3 5n) + (n 3 + 4n + 7) n 3 and n are different because the exponents are different 3n 3 + n 3 5n + 4n + 7 Group like terms 4n 3 1n + 7 To subtract polynomials add the opposite terms. For example: (4x 5) (2x + 2) 4x 5 + 2x 2 4x + 2x 5 2 6x 7 (5n 3 2m + 5) (2n 3 3m 1) 5n 3 2m + 5 + 2n 3 + 3m + 1 5n 3 + 2n 3 2m + 3m + 5 + 1 7n 3 + m + 6 Re-write with opposite terms Group like terms Re-write with opposite terms Group like terms
Assignment #3: Adding and Subtracting Polynomials Solve the following expressions. 1) (5p 2 3) + (2p 2 3p 3 ) 7) (5a + 4) (5a + 3) 2) (a 3 2a 2 ) (3a 2 4a 3 ) 8) (3x 4 3x) (3x 3x 4 ) 3) (4 + 2n 3 ) + (5n 3 + 2) 9) ( 4k 4 + 14 + 3k 2 ) + ( 3k 4 14k 2 8) 4) (4n 3n 3 ) (3n 3 + 3n) 10) (3 6n 5 8n 4 ) ( 6n 4 3n 8n 5 ) 5) (3a 2 + 1) (4 + 2a 2 ) 11) (12a 5 6a 10a 3 ) (10a 2a 5 14a 4 ) 6) (4r 3 + 3r 4 ) (r 4 5r 3 ) 12) (8n 3n 4 + 10n 2 ) (3n 2 + 11n 4 7)
13) ( x 4 + 13x 5 + 6x 3 ) + (6x 3 + 5x 5 + 7x 4 ) 20) (9r 3 + 5r 2 + 11r) + ( 2r 3 + 9r 8r 2 ) 14) (13n 2 + 11n 2n 4 ) + ( 13n 2 3n 6n 4 ) 21) ( 7x 5 + 14 2x) + (10x 4 + 7x + 5x 5 ) 15) (7 13x 3 11x) (2x 3 + 8 4x 5 ) 22) (13a 2 6a 5 2a) ( 10a 2 11a 5 + 9a) 16) (3y 5 + 8y 3 10y 2 ) ( 12y 5 + 4y 3 + 14y 2 ) 23) (8b 3 6 + 3b 4 ) (b 4 7b 3 3) 17) (k 4 3 3k 3 ) + ( 5k 4 + 6k 3 8k 5 ) 24) ( 7n 2 + 8n 4) ( 11n + 2 14n 2 ) 18) ( 10k 2 + 7k + 6k 4 ) + ( 14 4k 4 14k) 25) (14p 4 + 11p 2 9p 5 ) ( 14 + 5p 5 11p 2 ) 19) ( 9v 2 8u) + ( 2uv 2u 2 + v 2 ) 26) (8k + k 2 6) ( 10k + 7 2k 2 )
27) (4x 2 + 7x 3 y 2 ) ( 6x 2 7x 3 y 2 4x) (10x + 9x 2 ) 28) ( 5u 3 v 4 + 9u) + ( 5u 3 v 4 8u + 8u 2 v 2 ) + ( 8u 4 v 2 + 8u 3 v 4 ) 29) ( 9xy 3 9x 4 y 3 ) + (3xy 3 + 7y 4 8x 4 y 4 ) + (3x 4 y 3 + 2xy 3 ) 30) (y 3 7x 4 y 4 ) + ( 10x 4 y 3 + 6y 3 + 4x 4 y 4 ) (x 4 y 3 + 6x 4 y 4 )
Lesson #4: Distributive Property Monomial x Polynomial Distributive property allows you to expand an expression by multiplying the first term by each term in the polynomial. Remember the properties of exponents. For example: 3x(5x + 7) (3x)(5x) + (3x)(7) (3)(5)(x)(x) + (3)(7)(x) 15x 2 + 21x n(4m 5n + 3) (n)(4m) (n)(5n) + (n)(3) (4)(m)(n) (5)(n)(n) + (3)(n) 4mn 5n 2 + 3n
Assignment #4: Distributive Property Solve the following expressions. 1) 7(2n + 3m) 11) 9(3b 2) 2) 4(4r + 5h) 12) 6(j 1) 3) 6(3w + 2k) 13) 7(r 4) 4) 5(2q + 4) 14) (6k 2) 5) 8(2a + 1) 15) 8(g + h + 2r) 6) 9(2b 3) 16) 3(4a 3b 5c) 7) 3(3m 4) 17) 2( w 2 + 2w 5) 8) 4(p 2) 18) 10(0.4n + 0.2m) 9) 3(8e + 6) 19) 10(0.7n 2) 10) 5(5q + 2) 20) 100(0.03n + 0.25m)
Lesson #5: Multiplying Polynomials Polynomial x Polynomial To multiply two polynomials together, the word FOIL will help you remember each step. F O I L first outer inner last (6n + 3)(3n 4) (6n)(3n) = 18n 2 first (6n)(4) = 24n outer (3)(3n) = 9n inner (3)( 4) = 12 last 18n 2 + 24n + 9n 12 18n 2 + 33n 12 (3n 2 4)(2n 2 5) (3n 2 )(2n 2 ) = 6n 4 first (3n 2 )( 5) = 15n 2 outer ( 4)(2n 2 ) = 8n 2 inner ( 4)( 5) = 20 last 6n 4 15n 2 8n 2 + 20 6n4 23n 2 + 20
Assignment #5: Multiplying Polynomials Find the product for the following. 1) 6a(2a + 3) 7) (w 3)(6w 2) 2) 7( 5d 8) 8) (8a 2)(6a + 2) 3) 2v( 2v 3) 9) (6q + 8)(5q 8) 4) 4(g + 1) 10) (3h 1)(8h + 7) 5) (2r + 2)(6r + 1) 11) (2c 1)(8c 5) 6) (4m + 1)(2m + 6) 12) (5k + 6)(5k 5)
13) (4n 1) 2 19) (4q + 2)(6q 2 q + 2) 14) (7m 6)(5m + 6) 20) (7d 3)(d 2 2d + 7) 15) (6d + 3)(6d 4) 21) (7g 2 6g 6)(2g 4) 16) (8f + 1)(6f 3) 22) (y 2 + 6y 4)(2y 4) 17) (6w + 5)(5w + 5) 23) (6h 2 6h 5)(7h 2 + 6h 5) 18) (3y 4)(4y + 3) 35) (n 2 7n 6)(7n 2 3n 7) 24) (y + 5)(y 2) 25) (g 1)(g + 1)
26) (q 1) 2 33) (8m 2 + 4)(8m 2 4) 27) (y 3)(y + 3) 34) (2 + 5m 2 ) 2 28) (y 4) 2 35) (3y 7)(3y + 7) 29) (m + 3) 2 36) (3 + 7x 2 )(3 7x 2 ) 30) (y 5)(y + 5) 37) (7x 2 6)(7x 2 + 6) 31) (a 5) 2 38) (2 + b) 2 32) (2b 2 + 1) 2 39) (6x + 3)(6x 3)