Factor each expression. Remember, always find the GCF first. Then if applicable use the x-box method and also look for difference of squares.
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1 NOTES 11: RATIONAL EXPRESSIONS AND EQUATIONS Name: Date: Period: Mrs. Nguyen s Initial: LESSON 11.1 SIMPLIFYING RATIONAL EXPRESSIONS Lesson Preview Review Factoring Skills and Simplifying Fractions Factor each expression. Remember, always find the GCF first. Then if applicable use the x-box method and also look for difference of squares..) Check: Check: 3.) 4.) Check: Check: 5.) Check: Check: Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 1
2 Simplify (if possible) each fraction Simplifying Rational Expressions Key Terms: Simplify: Example: Simplify 1 15 = Rational Expression: Example: 3 6xy x4,, xy x 4 Negative Rule: Helps us to simplify opposites in the numerator and denominator. example: simplify Practice: rewrite each term using the negative rule (factor out a.) Simplifying Rational Expressions: Factor all expressions (numerators and denominators) [Make sure you have factored correctly by!!!] Reduce by canceling out (terms in parentheses) from the numerator and denominator and don t forget to look for ways to apply the negative rule Re-write answer as a new and reduced fraction Mrs. Nguyen Algebra 1B Chapter 11 Notes Page
3 Simplify: 4.) 4( x 3) 10( x 3) 5.) 6( x 1)( x+ 5) 15(1 x)( x+ 3) 6.) 6x 18 5x 15 7.) 4 x x x 8.) x x3 x 1 9.) m m + 5m m ) w + w 4 w 5w ) 3 y y y y + 3y+ 1.) 3 8k k 3 5k 0 13.) h + 9 h + 3 Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 3
4 LESSON 11. MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS Factoring Review and Factoring Perfect-square trinomials Take out GCF Factor using Box (4-terms) or X-Box ( or 3 terms) Check using FOIL If the answer is the product of two identical binomials, write as ( ) Factor. 1.) x 8x + 16.) n + 16n ) 3m 18m+ 7 4.) 6x 54 5.) 3n n ) 9g + 1g ) 4t + 36t ) 3 y y y ) 3a 6a ) x ) 4w 49 1.) 4x ) 9v x 40 15) 3c 75 Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 4
5 Lesson Preview: Multiply the fractions. Leave your answer in simplest form. Multiplying Rational Expressions: Factor all expressions (numerators and denominators) [Make sure you have factored correctly by!!!] Reduce by canceling out (terms in parentheses) from the numerator and denominator [Look for ways to!!!] Re-write answer as a new and reduced fraction Multiply: 1.) ( x 3) 9( x+ 1) i.) 3( x+ 1) 4( x+ 3) 6( x 1) ( x 5) i + 9( x + 5) 4(1 x) 3.) 6x1 3x9 5x15 x4 i 4.) x 4 x 1 i 3 + x x x Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 5
6 5y y+ 6 y 9 15y 5.) i 3 6.) 3x 4 4 i x 4 3x 7.) c 5c3 c d c+ d c+ 1 i 8.) 3 y + y 4y 8 y 1 i y + 3y+ 3y+ y 9.) 3( k 3) ( x+ ) 4x+ 8 6k i 10.) 4 ( h ) h h 1 i ( h+ 1) h 4h+ 4 Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 6
7 Lesson Preview: Dividing Fractions To divide fractions, the second fraction over and make it a problem. Divide the fractions. Leave your answer in simplest form. 11.) 8 1.) xy 7xy ab ab Dividing Rational Expressions: Factor all expressions (numerators and denominators) [Make sure you have factored correctly by!!!] the second fraction over and make it a problem. Reduce by canceling out (terms in parentheses) from the numerator and denominator [Look for ways to!!!] Re-write answer as a new and reduced fraction Divide: 13.) 4( x 5) (5 x) 9( x+ 1) 3( x+ 3) 14.) 3( x 3) ( x+ 4) 9( x + 5) 4(5 x) Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 7
8 15.) 3x1 5x0 x0 x4 16.) x 8 x x x x ) 3 6y 1y y y y 18.) m 6 m 3 4m+ 8 m ) c c + c 3 6c 6 3c + 9c 0.) y y y y y y y Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 8
9 LESSON 11.3 DIVIDING POLYNOMIALS Dividing Polynomials by a Monomial Re-write the large fraction as many fractions; each piece of the written over the. Reduce each fraction ) 14x y + 8x y 6xy xy ) 18rt 1rt 6rt 3 3rt ) 14ab + 1ab 8ab 3 7ab ) 5x y + 0x y 16x y 3 3 5xy Dividing Polynomials by a Binomial Lesson Preview: Review Long Division Process: Divide Multiply Subtract Bring Down Repeat as Necessary Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 9
10 Polynomial long division follows the same steps, but there are more pieces to keep track of: (x + 6x 10) (x + ) 1.) put expression into the division sign.) times what makes? Put this number on 3.) this number to the outside, put below 4.) ; change to addition and distribute the subtraction sign. 5.) Add, bring down next number. 6.) Steps 5 again 7.) Put remainder over the divisor. Divide: 8.) (3x 3 + x 4x) (x) 9.) (x 3 5x + 4x ) (x ) Things to watch out for: The dividend (the expression inside the division sign) and the divisor must be in order If either the dividend or the divisor is missing pieces (like there is no x term ) you must fill in the missing pieces, like 10.) (a 3 + a 3) (a + ) 11.) (15y 3 + y 1y) (5y 3) 1.) (x 3 + 1) (x + 1) 13.) (6x 4 + 5x 3x 3 + x 6) (3x ) Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 10
11 LESSON 11.4 ADDING AND SUBTRACTING RATIONAL EXPRESSIONS Lesson Preview: Adding and subtracting fractions When adding/subtracting fractions, you need a Add/Subtract the numerators. Leave the denominators Factor and reduce Find the LCM (Least Common Multiple) of each pair of numbers: 4.), 6 5.) 4, 6 6.) 1, 16 Add or subtract the fractions Add and subtract rational expressions with like denominators 1.) 1 6z 6z +.) 3bx cx ax ax 3.) 5 3 6x y 6xy x x+ 3 x 9 x 9 4.) 5.) m 5 + 3m m3 6.) a 3a a+ a+ Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 11
12 Add and subtract rational expressions with unlike denominators Find the LCM (Least Common Multiple) of each pair of polynomials: Factor each polynomial The LCM is the collection of the factors in those polynomials 7.) 3a 6, 4a 8 8.) x 9, x x 3 9.) 8y 1y 8, 4y + 6y + Add/Subtract. Simplify. each polynomial Find the (LCM) Multiply top and bottom of each fraction by what it is missing Add/Subtract the numerators. Leave the denominators Factor and reduce 10.) 5 6z 3z + 11.) 3 + ax ay 1.) 5 3 6x y 9x3y 13.) x+ 6 x+ 3 + x+ 3 x ) 1 1 m + m+ 1 m x + x6 x 3x+ 15.) Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 1
13 LESSON 11.5 SOLVING RATIONAL EQUATIONS Solve: Factor denominators, find Set LCM = 0 and solve; these are the restrictions on the domain Multiply top and bottom of each term by what is missing from the LCD to make all denominators match; look for when to Butterfly and/or BOX! Cancel out denominators; solve the left-over equation If you are left with a quadratic; solve by,, or Compare your solutions to the restrictions 1.) =.) x 5 6 x + = 5 x LCM = x LCM = x 3.) = 1 x 4 x+ 4 4.) 3 5 = x x = 1 x+ 1 x x x 5.) = 4 y y+ y 4 6.) Work/Job Problems 7.) If you can paint a fence in 4 hours, what fraction of the job can you do in 1 hour? Meaning, what is your rate? Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 13
14 8.) If you can write a paper in3 hours what fraction of the paper can you write in 1 hour? Meaning, what is your rate? 9.) If you can solve 1 math problems in one hour, how many can you solve in 3 hours? Let s generalize this relationship into a formula: Work Problems: Fill in x = across the top of your chart Let t = time Rate = 1 hours it takes to complete a job Add jobs to = 1 whole job 10.) Laura can paint a fence in hrs. It takes her brother 3 hrs to paint the same size fence. If they worked together, how long would it take them to paint the fence? Laura Her brother Laura s job + her brother s job = 1 painted fence 11.) Christian can write his English essay in 3 hours. It takes his partner 4 hours to write it. If they worked together, how long would it take them to write an essay? Christian Partner 1.) Garden Hose A can fill a jacuzzi in 10 hours. Garden Hose B can fill a Jacuzzi in 1 hours. How long would it take both hoses to fill the tank? Hose A Hose B Mrs. Nguyen Algebra 1B Chapter 11 Notes Page 14
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