Name: Directions: Use pencil and the space provided next to the question to show all work. The purpose of this packet is to give you a review of basic skills. Please refrain from using a calculator! Prepared by: Mrs. Trebat Fairfield Ludlowe High School mtrebat@fairfield.k1.ct.us 1
BRUSHING UP ON BASIC ALGEBRA SKILLS Mrs. Trebat Name: DUE DATE: Directions: Use pencil, show work, box in your answers. Monomial Factors of Polynomials A monomial is an expression that is either a numeral, a variable, or the product of a 3 numeral and one or more variables. Example of monomials: 7, x, 6x y. A sum of monomials is called a polynomial. Some polynomials have special names: Binomials (two terms): 3x 5 or xy x Trinomials (3 terms): x 5x 15 or x 6xy 9y Divide: 4x 1 1) 6 ) 4 3 8x 4x 6x x Factor (monomials only): Example: 15a 5b 35 5(3a 5b 7) 3) 3 7 1 14 a a a 4) 3 5ax 10a x 15a Simplify: Example: 14x 1 10x 5 7(x 3) 5(x 5) x 3 x 5 7 5 7 5 Factor 5) 6 a 9 b 7 a 1 b = 3 7 Cancel out common factors Simplify and combine 6) x y 3x y 6xy 9xy xy 3y
Multiplying Binomials Mentally Write each product as a trinomial: 7) ( x 9)( x 4) 8) (4 x)(1 x) = 9) (a 5)( a ) 10) (x 5)(3 x 4) Find the values of p, q, and r that make the equation true. 11) ( )( 5) 6 11 px q x x x r Difference of Two Squares You must use the shortcut below (do not FOIL!!!!) ( a b)( a b) a b ( First Second ) ( First Second ) First Second Write each product as a binomial: 1) ( x 7)( x 7) 13) ( y 8)( y 8) 14) (5x )(5x ) = 15) 8x 118x 11 16) 4a 5b 4a 5b 17) x 9y x 9y Now let s try reversing the process above Factor: 18) x 36 19) m 81 0) 5a 1 = 1) 49x 9y Factor each expression as the difference of two squares. Then simplify. Example: x x 3 x x 3 x x 3 3x 3 ) Apply the formula x 4 x 3) simplify 9( x 1) 4( 1) x = 3
Squares of Binomials and Perfect Square Trinomials Every time you square a binomial, the same pattern comes up. To speed up the process, we should memorize: a b a ab b and a b a ab b A trinomials is called a perfect square trinomial if it is the square of a binomial. For example, x 6x 9 is a perfect square trinomial because it is equal to x 3. Write each square as a trinomial. 4) a 9 = 5) x 7 6) 4x 1 7) 5a b Decide if each trinomial is a perfect square. If it is, factor it. Otherwise, write not a perfect square. 8) a 6a 9 9) y 14y 49 30) 11 x x 31) 9a 30ab 100b 3) 49x 8xy 4y 33) 5x 15xy 36y = 34) a b 1ab 36 35) 4 11 33x 9x 36) Show that a 8a 16 can be factored as ( a ) a. 4 37) Solve and check: x x 3 35 4
Factoring Quadratic Trinomials To factor a trinomial of the form x bx c, you must find two numbers, r and s, whose product is c and whose sum is b. x bx c x r x s When you find the product x r x s you obtain x bx c x ( r s ) x rs Example: Factor x x 15. a. List the factors of -15 (the last term). Factors Sum of the b. Either write them down or do this mentally. of -15 factors Find the pairs of factors with sum - (the middle term). 1, -15-15 (discard) -3, 5 (discard) 3, -5 - (keep) x x 15 ( x 5)( x 3) Check the result by multiplying Factor. Check by multiplying (mentally). 38. x 8x 1 39. x 7x 1 40. x 4x 1 41. x 9x 18 4. x 3x 18 43. x 11x 18 44. x 5x 36 45. x 15x 36 46. x 9x 36 47. x 3x 8 5
Factoring General Quadratic Trinomials of the type ax bx c Example: Factor = (x 9)( x 1) x 7x 9 Factor: List all factors List factors of -9 Of x 48) 3x 7x 49) x 5x 3 50) x 15x 7 51) 3a 4a 4 5) 5a 6a 53) 3x x 5 54) 3m 7m 6 55) 4a a 3 Factor by Grouping Example 1: 7( a ) 3 a( a ) (7 3 a)( a ) Notice that we factored out the common factor (a - ) Example : 5( x 3) x(3 x) Notice that x 3 and 3 x are opposites. 5( x 3) x(3 x) 5( x 3) x( x 3) (5 x)( x 3) 56) x ( x y ) y ( y x ) 57) 3 a( b a) b( a b) Group and Factor: 58) 3a ab 3c bc 59) 3 3a a 6a 60) 3ab b 4 1a 6
Solving Equations by Factoring The Zero Product Property Key Concept: Zero-Product Property For all real numbers a and b: a b 0 if and only if a 0 or b 0 A product of factors is equal to zero if and only if at least one of the factors is 0. Solve: 61) ( x 5)( x 3) 0 6) xx ( 9) 0 63) (a 3)(3 a ) 0 64) 3 x(5x )( x 7) 0 65) a 3a 0 66) x 1x 35 0 67) b 4b 3 68) 5x 16 0 69) 7x 18x 11 70) 3 8y y 0 71) 3 4x 1x 8x 0 7) 3 9x 5x 30x Sample for items 73-74: ( a 1)( a 3) 1 a a 3 1 0 expand the left side; bring over 1; set it = 0; ( a 3)( a 5) 0 combine like terms; factor it; a 3or a 5 73) ( x 1)( x 5) 16 74) (z 5)( z 1) 7
Simplifying Fractions Follow the examples below. Example 1: Simplify 3 x 6. 3x 3y 3x 6 3( x ) Solution: 3x 3y 3( x y ) x, x y x y Factor the numerator and denominator; look for common factors; Cancel out common factor which is 3; x 9 Example : Simplify (x 1)(3 x) Solution: x 9 ( x 3)( x 3) (x 1)(3 x ) (x 1)( x 3) x 3, x 1, x 3 x 1 First factor the numerator; pull out a negative in the factor (3-x) to make it a common factor with the numerator; exclude the first as it would make the denominator =0; exclude the nd as it would make both numerator and denominator =0. Simplify. Give any restrictions on the variable. 75) 5 x 10 x a 4 76) a 4 x 77) 5 7 x (5 x)(7x ) 78) x 8x 16 16 x 8
Multiplying Fractions Multiplication Rule for Fractions a c ac To multiply fractions, you multiply their numerators and multiply their denominators. b d bd Note: You can multiply first and then simplify, or you can simplify first and then multiply. Example: x x 1 x 5 x 4 x 3 x 5 x 5 x 5x x 3 x x 5 x 3 = x 4 x 5 x Notice how common factors were cancelled. Simplify. a 3a 79) a a 4 80) a x a a 3x 3a 81) A triangle has base 3 x 8 cm and height cm. What is its area? 4 9x Dividing Fractions a c a d 7 7 3 7 Division Rule for Fractions: Example : b d b c 9 3 9 6 To divide by a fraction, you multiply by its reciprocal! 9
8) a 1 a 83) 6 9 x 1 x 1 16 84) a b a b 85) a 4a 4x 5 1x 30 x 16 x 8x Adding and Subtracting Algebraic Fractions Key Steps: (1) Find the Least Common Denominator (LCD); () Re-write each fraction being added or subtracted with the same common denominator. (3) Add or subtract their numerators and write the result over the common denominator. Example: 3 8 3 8 6x 30 9x 45 6( x 5) 9( x 5) 9 16 18( x 5) 18( x 5) 5 5 = or 18( x 5) 18x 90 86) 4 x 3 7 x x 3 3 4 6 Factor out denominators to more easily identify LCD multiply the first fraction by 3, the second by 10
87) 4 x 3 x 3 = x 4 88) x 1 x 1 89) 3a 6b a b b a x 11 x 7 90) = x 9 x 3x 11
BRUSHING UP ON BASIC ALGEBRA SKIUS - '501ue4 Mn. Treht Name:.- birections: Use pencil: You must show work for credit. Your fim/ansners must be clear& identified Monomial Factors of Polynomials A monomial is an expression that is either a numeral, a variable, or the product of a numeral and one or more variables. Example of monomials: 7, x, 6xy3. A sum of monomials is called a polynomial. Some polynomials have speaal names: Binomials (two terms): 3x - 5 or xy + x Trinomials (3 terms): xt + 5x - 15 or xt - 6xy + 9y Divide: Multiptyinq Binomials Mentally Write each product as a trinomial: 7)(x-9)(~+4)= xz-sx-36 8)(4-x)(1-x)= L f - 5 ~ t n ~ w Find the values of p, q, and r that make the equation true. 11) (px + q)(x + 5) = 6x + 11x + r apx+ (cp+nq>r +5q = brztlln tr dp?+spx +qx+ 57 = ~~uafi'ny I'-lc= rerrr': bifference of Two Squares p z,a ~ 5?+p=li 5(&)=,y+Zqzll \-IO-r Write each product as a binomial: 1) (x+7)(x-7) = x- yq 13) (y+8)(y-8)= y-67 Factor: Example: 15a - 56 + 35 = 5(3a - 56 + 7) - Simplify: Example: Now let's try reversing the process above... Factor: 5) Factor Cancel out common factors Simplify ond combine 6a+9b 7a+lb- j/(zai3b) - j (e+3b), at3b-a-3b=a 3 7 P f Factor each expression as the difference of two squares. Then simplify. Apply the formula simplify
Sauares of Binomials and Perfect Square Trinomials Every time you square a binomial, the same pattern comes up. To speed up the A trinomials is called a perfect square trinomial if it is the square of a binomial. For example, x - 6 x + 9 is a perfect square trinomial because it is equal to (x - 3y. Write euch square as a trinomial. 4) (a- 9) = az- I~L + d I 5) (x+7)= y+1lllc+ll(l Factorinq Quadratic Trinomials To factor a trinomial of the form x + bx + c, you must find two numbers, r and s, whose product is c and whose sum is b. x+bx+c=(x+r)(x+s) When you find the product (x + r)(x + s) you obtain Example: Factor x - x - 15. a. List the factors of -15 (the last term). Factors Sumof the b. Either write them down or do this mentally. of -15 factors Find the pairs of factors with sum - (the middle term). *discard) 1, -15-15 (discard) Decide if each trinomial is a perfect square. If it is, factor it. Otherwise, write not a perfect square. 8) az + 6a + 9 = (a+ 3)' 9) y-14y+49= t 30) 11-x+x =_ ([I-K) 31) 9a + 30ab + 100bz = hlot A PtwkzrSQ+~~fc ( due +D middle Tunl.,, ) :.xz-x-15=(x-5)(x+3) Check the result by multiplying... Factor. Check by multiplying (mentally). 3, -5 - (keep) 36) Show that a4-8az + 16 can be factored as (a + )'(a -). a'- ~ o - ~ t l b = (az-'1)' = C&+~)(a-r)]~ = (G+~~(G-~I~ 37) Solve and check: (x + ) -(x- 3)t = 35 - (4-i-br +q) =35- qr+~+dr-~i = 33- LOX -5 = 3s 1 0 = ~ (lo
m &La C + M O O L h rn
Simplifiinq Fractions - Follow the examples below. 3x + 6 / Example 1: Simplify - i Solution: 3x+3y' 3x -- 3(x 3x + 3y 3(x + y), Factor the numerator and demmimtor look for common factors: 1 x + j --, X # -y Cnncel out common factor which is 3; x + Y xz-9 Example : Simplify (x + 1)(3 - x) Solution: xz-9 = (x + 3)(x - 3) First factor the numerutor; 'pull out' a (zx + 1)(3 - x) -(zx + l)(x - 3) "Wve in the fattor (3-X) to mk it a common factor with the numerator: x + 3 exclude the first m it would make the denomimator =O; uci& the '"'' it would mke both numerator and denomi~tor =O. I b Multiplyinq Fractions 1 Multiplication Rule for Fractions I -.-=a c ac d bd To multiply fractions, you multiply their numerators and multiply their denominators. Note: You can multiply first and then simplify, or you can simplify first and then multiply. Example: x - x - 1.-= xz - 5 (x - 4)(x/) (&)(x + 5) xz-5x x+3 Simplify. (x -4)(x + 5) Notice how common factors were cancelled. X Simplify. Give any restrictions on the variable. 3x 8 81) A triangle has base -cm 4 and height -cm. 9x What is its area?,, & ~&wc)[he~hi) L Dividi~ Fractions