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Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Algebra 1 Unit 6: Polynomials Lesson 1 (PH Text 7.1): Zero and Negative Exponents Lesson 2 (PH Text7.2): Scientific Notation Lesson 3 (PH Text 7.3): Multiplying Powers with the Same Base Lesson 4 (PH Text 7.4): More Multiplication Properties of Exponents Lesson 5 (PH Text 7.5): Division Properties of Exponents Lesson 6 (PH Text 8.1): Adding and Subtracting Polynomials Lesson 7 (PH Text 8.2): Multiplying a Polynomial by a Monomial Lesson 8 (PH Text 8.2): Monomial Factors of Polynomials Lesson 9 (PH Text 8.3): Multiplying Binomials Lesson 10 (PH Text 8.4): Multiplying Polynomials: Special Cases Lesson 11 (PH Text 8.5): Factoring x2 + bx + c, c > 0 Lesson 12 (PH Text 8.5): Factoring x2 + bx + c, c < 0 Lesson 13 (PH Text 8.6): Factoring ax2 + bx + c Lesson 14 (PH Text 8.7): Factoring Special Cases Lesson 15 (PH Text 8.8): Factoring by Grouping Lesson 16 (PH Text 11.1): Simplifying Rational Expressions 1 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 1 (PH Text 7.1): Zero and Negative Exponents Objective: to simplify expressions involving zero and negative exponents Properties: Zero as an Exponent – For every nonzero number a, a0 = 1. Examples: 80 = (-4)0 = (3.14)0 = 1 Negative Exponent – For every nonzero number a and integer n, a n n . a Examples: 8-2 = (-4)-3 = (3.14)-2 = Discussion: What about 00? What about 9x0? Class Practice: 1) 3-4 = 2) (7.89)0 = 6) 8x3y-2 = 8) 10) 1 4 3 4b 2 3 2 a 3) (2.5)-3 = 4) (-16)-2 = 7) 3-2x-9y5 9) 1 an 11) HW: p.417 #8-58 even 2 7m0 n5 p 1 5) 2-1 = Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 2 (PH Text7.2): Scientific Notation Objectives: to write numbers in scientific and standard notation to compare and order numbers using scientific notation Complete the table. Notice the pattern. 103 = = 10 2 = = 101 = = 100 = = 101 = = 102 = = 103 = = Scientific Notation – a number expressed in the form a x 10n , where n is an integer and 1 ≤ |a| < 10. Examples: 1) Is the number written in scientific notation? Explain why/why not. a) 2.36 x 104 b) 762.1 x 10-3 c) 0.41 x 10-8 2) Find each value. a) 2.36 x 104 = b) 7.1 x 10-3 = Shortcut hint: 3) Write each number in scientific notation. a) 18,459 = b) 0.00987 = Shortcut hint: 3 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Comparing Numbers in Scientific Notation: First compare the powers of ten. If the numbers have the same power of 10, then compare the front parts. Example: Write the numbers in order from least to greatest. a) 1.23 x 107, 4.56 x 10-3, 7.89 x 103 b) 0.987 x 103, 654 x 103, 32.1 x 103 11. 3(4 x 105) _________________ 12. 2(7 x 102) _________________ 13. 10(8.2 x 1012) _________________ 14. 6(3 x 108) _________________ HW: p.423 #9, 12-46 even 4 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 3 (PH Text 7.3): Multiplying Powers with the Same Base Objectives: to identify monomials to multiply monomials with the same base Monomial - a real number, a variable, or a product of a real number and one or more variables Examples: 1) Determine whether or not the given term is a monomial. 1 13 a) x 2 yz 2 b) c) 7g 23 2 2 8a 2 b 5 d) c Property: Multiplying Powers with the Same Base When multiplying monomials with the same base, ADD the exponents: a m a n a mn a3 a5 (a a a) (a a a a a) a35 2 3 2 2 3 21 2 31 2 4 Examples: 2) Write each expression using each base only once. a) 23 · 25 · 2-4 b) (0.6)-9(0.6)-8 3) Simplify a) a 2 b 3 a 2 d) 2 x 2 y 3x 5 y 2 b) 2 x 3x 3 4 x 2 g) 0.5 1013 0.3 104 e) 3m 2 n 5 8mn 2 c) 2r 3 x r 2 x d x h) 4 10 6 3 10 3 HW: p. 429 #7, 9-63 multiples of three 5 f) 3103 7 109 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 4 (PH Text 7.4): More Multiplication Properties of Exponents Objectives: to raise a power to a power to raise a product to a power Monomial(s) Raised to a Power: When a monomial is raised to a power, you multiply the exponents (a m )n a mn (ab)m a mbm Examples: 1) Simplify: a) (2 2 ) 3 b) [(2) 2 ]3 c) (2 2 ) 3 d) (x 6 9 ) e) (ab) 3 f) (8x5)3 j) (2a 2 ) 3 (3b) 2 2 i) (-7 x 105)2 HW: p.436 #8-54 even (GOOD IDEA: Mid-Chapter Quiz p.439) 6 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 5 (PH Text 7.5): Division Properties of Exponents Objectives: to divide monomials to raise a quotient to a power Property of Exponents for Division When dividing monomials with the same base, SUBTRACT the exponents am a mn n a a 5 (a a a a a) a 53 3 a (a a a) 144 2 2 2 2 3 3 2 4 32 3 2 2 43 322 72 2 2 2 3 3 2 3 Examples: 1) Simplify: a) y6 y3 b) y2 y7 c) 6 1012 f) 2 10 6 12k 2 m 3 n e) 9m 3 n 6 k 5 4a 2 b 5 d) 2a 6 b 2 y 16 y 16 6 x 3 y 2 2 g) 9 x y 2) Find the value of x in each equation: 3x 1 a) 9 3 3 3 Simplify: 4 10 3 10 6 e) f) 3 9 10-12 -3 109 t 5 x b) x t 3 t c 4 g) d 2c h) 2 d 7 4 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials (2a 2 )3 i) 2 (3a) 2 a 2 m3 j) m 1 a 2 HW: p.443 #8-52 even, 70, 80 If lessons 4 and 5 are combined… HW: p.437 #22-36 even, 44-54 even; p.444 #21-24, 32-52 even, 70, 80 8 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 9 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 6 (PH Text 8.1): Adding and Subtracting Polynomials Objectives: to classify, add, and subtract polynomials Vocabulary: Descending order - writing the order of the variables from highest power to lowest power Ascending order - writing the order of the variables from lowest power to highest power Monomial - has one term; example: 0.006t Binomial - has two terms connected by addition or subtraction; example: 3x + 2 Trinomial - has three terms connected by addition or subtraction; example: 3x 2 2 x 1 Polynomial – is a monomial or a sum or difference of monomials Degree of a term - exponent of the variable (each monomial is a term) Degree of a polynomial – is the highest degree of any of its terms after it has been simplified Polynomial Degree Name Using Degree Number Of Terms Name Using Number of Terms 7x 4 3x 2 2 xy 2 1 4x 3 y4z 5 x3 x 2 x 1 Polynomials can be simplified by combining like terms. Examples: 1) State the degree: 1 a) x 2 b) 8a 2 b 5 2 2 x y 8 xy 2 5 3 3) Simplify: 3r 2 s 2 5rs 3 9r 2 s 2 4s = 2) State the degree of 2 x 2 y 2 4) Simplify: 0.3xy 2 0.9 x3 y 0.3xy 2 0.6 x3 y 2.4 = 10 c) 6 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Standard form - terms are in alphabetical order - terms decrease in degree from left to right - no terms have the same degree (when more than one variable, with respect to the first variable in the alphabet) Write each polynomial in standard form, then name each by its degree and number of terms 1) 2 7x 2) 2 x3 x 4 3 3) 3x 4 2 2 x 4 7 x 4) 6 x 2 7 9 x3 Some algebraic expressions are not polynomials Polynomial x 4 x 2 3 1 2 x 3 3 2 y x2 z 8 3 x2 Why it is not a Polynomial 11 An algebraic expressions is NOT a polynomial if it: 1) has a negative exponent 2) is not a sum or difference 3) has a variable in the denominator 4) has more than one variable Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Adding (two options): 1. 4 x 3 x 2 3x 5 x 2 4x 6 + 4 x3 x 2 3 x 5 vertical b. horizontal 4x3 x 2 3x 5 x 2 4x 6 Subtracting (two options): 2. x2 4x 6 a. align like terms BE CAREFUL! 3 2 4a 3a 3a 5 – 2 a 2a 7 4a3 3a 2 3a 5 a 2 2a 7 a. vertical b. horizontal (add the opposite) Distribute the negative! → align like terms 4a 3 3a 2 3a 5 a 2 2a 7 3 2 2 4a 3a 3a 5 a 2a 7 12 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 3 8n 7n 4 3. – 3 2 3n 2n 7 8n3 a. vertical b. horizontal 7n 4 3n 3 2n2 7 align like terms Distribute the negative! 8n3 7n 4 3n3 2n 2 7 HW: p.477 #9-27 multiples of 3, 30-40, 44-48 13 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 7 (PH Text 8.2): Multiplying a Polynomial by a Monomial Objectives: to multiply a polynomial by a monomial to simplify algebraic expressions that involve multiplication of a polynomial by a monomial Use the Distributive Property: 1. 5x2 y 3x 2. -6x(x2 – xy + y) 3. 2a 2b 5ab2 3b 4a 7 4. 4 x 2 2 x 1 3x6 x 4 5. 6ab2 2a 2b 3a 5 3a 2b 4ab2 2b 14 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials GCF and LCM with Variables Objectives: to find the greatest common factor and the least common multiple of a set of monomials Greatest Common Factor (GCF) Least Common Multiple (LCM) Find the GCF of 32 and 24. Method 1 – “Rainbow Method” List all factors of 32 and 24. 32 – 1, 2, 4, 8, 16, 32 Method 2 – Prime Factorization List the prime factors of 32 and 24. 32 – 25 24 – 1, 2, 3, 4, 6, 8, 12, 24 24 – 23·3 Common factors: 1, 2, 4, 8 common prime factor is 2 GCF = 8 lesser power of that prime factor is 23 GCF = 23 = 8 Method 3 – Ladder Method 2 32 24 Is there a common factor? 2 16 12 yes 2 8 6 yes 4 3 no ↑ GCF = 2·2·2 = 8 for LCM, “use the “L” LCM = 2·2·2·4·3 = 96 Find the GCF of 36m3 and 45m8. Method 1 List all factors of 36m3 and 45m8. 36m3 – 1, 2, 3, 4, 6, 9, 18, 36 · m· m· m Method 2 List the prime factors of 36m3 and 45m8. 36m3 – 22·32· m3 45m8 – 1, 3, 5, 9, 15, 45 · m· m· m· m· m· m· m· m 45m8 – 32·5· m8 Common factors: 1, 3, 9 GCF = 9m3 · m· m· m common prime factor is 3 and m lesser power of that prime factor is 32 and m3 GCF = 32 · m3 = 9m3 15 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Find the GCF of 36m3 and 45m8 using the Ladder method. 3 36m3 45m8 Is there a common factor? 3 12 m3 15 m8 yes m 4 m3 5 m8 yes m 4 m2 5 m7 yes m 4m 5 m6 yes 4 5 m5 no ↑ GCF = 3·3·m·m·m = 9m3 for LCM, “use the “L” LCM = 3·3·m·m·m·4· 5m5 = 180m8 Practice: Find the GCF. 1. 60x4 and 17x2 _______________ 2. 32y12 and 36y8 _______________ 3. 16n3 , 28n2 and 32n5 _______________ 4. 16m10 , 18m and 30m3 _______________ Find the LCM. 5. 60x4 and 17x2 _______________ 6. 32y12 and 36y8 _______________ 7. 16n3 , 28n2 and 32n5 _______________ 8. 16m10 , 18m and 30m3 _______________ Review: Multiply. 11. 4(x2 + 3x + 2) _______________ 12. a(a + 7) _______________ 13. 2p(p2 + 2p + 1) _______________ 14. 3xy(z2 + 6z + 8) _______________ HW: p.482 #5-20 16 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 8 (PH Text 8.2): Monomial Factors of Polynomials Objectives: to factor the greatest common monomial factor from a polynomial To factor a polynomial: 1 – Find the GCF of the terms. 2 – Use the distributive property (in reverse). 3 – … more to follow in future lessons ... Sample: 1. 10xy – 15x2 ← Find the GCF of 10xy and 15x2 5x(2y – 3x) ← Use the GCF, and what remains of each term with the distributive property. 2. 10a 3b3 6a 2b 2 8a 9b 3. 8x 3 14 x 2 12 x 4. 12a 2b 2 30ab2 17 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Geometry Application: Reminder: Area of a Circle: The area of a circle is the product of and the square of the radius. A r2 The rectangle has sides measuring 4 cm and 6cm. Find the area of the shaded region. HW: p.483 #21-28, 36 18 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 9 (PH Text 8.3): Multiplying Binomials Objectives: to multiply two binomials, or a binomial and a trinomial To multiply two binomials: “Double” Distribute Method 1. a 4a 5 aa 5 4a 5 Table Method (x – 7)(2x + 9) Make a table of products. 2x 9 Write out all of the product terms and simplify. x -7 FOIL Method (shortcut to other methods) 2. s 6s 3 3. 2x 3x 5 F O I L – – – – First terms Outer terms Inner Terms Last Terms F O I L s s s (3) 6 s 6 (3) F O I L 2x x 2x (5) 3 x 3 (5) 19 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 4. 4a b2a 3b 5. 8 j 32 8 j 38 j 3 To multiply any two polynomials 6. 2t 5 3t 4t 2 2t 3 “Double” Distribute Method (Horizontal) a) 2t 3t 4t 2 2t 3 5 3t 4t 2 2t 3 Arrange in descending order method (Vertical) 2t 3 4t 2 3t 2t 5 b) 20 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Multiply the polynomials: 7. a 2 3a 9 a 4 (Solve using both methods.) “Double” Distribute Method (Horizontal) a) a 2 3a 9 a a 2 3a 9 4 Arrange in descending order method (Vertical) a 2 3a 9 a4 b) HW: p.489 #13-14, 27-28, 30-42 even 21 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Extra practice: 22 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 23 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 10 (PH Text 8.4): Multiplying Polynomials: Special Cases Objectives: To find the square of a binomial To find the product of the sum and difference of two terms Product of the sum and difference of same two terms: m nm n = m2 mn mn n2 = m 2 n 2 STEPS 1. square the first term second term 2. square the 3. write the difference of the two squares Examples: 1. x 5x 5 2. 4x y 4x y Square of a binomial m n2 = m nm n = m2 mn nm n2 = m 2 2mn n 2 or m n2 = m nm n = m2 mn nm n2 = m 2 2mn n 2 STEPS 1. square the firstterm 2. double the product of the two terms 3. square the second term 4. write the sum of the three new terms Examples: 3. 4x 2 y 2 m 4x n 2y 4. 3a 4b2 m= n= 24 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 2 2x 2 3 5. 6. 2 5x 2 2 7. (23)2 = (20 + 3)2 8. (41)2 = ( 9. 3x 2 y 3x 2 y )2 HW: p.496 #17, 26-52 even 25 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials MID-CHAPTER QUIZ: p.498 26 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 11 (PH Text 8.5): Factoring x2 + bx + c, c > 0 Objectives: to factor a trinomial of the form x2 + bx + c, c > 0 Factoring: Find two binomials that will multiply to be the quadratic expression given --- FOIL backwards. 1. Draw the parentheses. ( )( ) 2. Put two first terms in the ( ) that will multiply to be the first term of the quadratic. 3. Find two second terms for the ( ) that will multiply to be the last term of the quadratic, but add to be the middle term of the quadratic. s 6s 3 1) x2 + 5x + 6 F O I L s s s (3) 6 s 6 (3) 2) x2 – 13x + 12 3) (x – 1)(x – __) (x + 2)(x + __) x2 – 18x + 17 (x – __)(x – __) 4) x2 + 4x + 3 5) x2 + 3x + 2 6) x2 – 6x + 5 7) 11 – 12p + p2 8) 7 + 8m + m2 9) d2 – 8d + 12 10) 21 – 10p + p2 11) 27 + 12x + x2 12) d2 – 9d + 14 13) x2 – 10x + 25 14) x2 + 12x + 32 15) x2 + 16x + 48 HW: p.503 #10-19 27 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 28 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 12 (PH Text 8.5): Factoring x2 + bx + c, c < 0 Objectives: to factor a trinomial of the form x2 + bx + c, c < 0 HW: p.503 #20-44 even 29 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 13 (PH Text 8.6): Factoring ax2 + bx + c Objectives: to factor a trinomial of the form ax2 + bx + c The process of factoring a trinomial is finding two binomials whose product is the given trinomial. Basically, we are reversing the FOIL method to get our factored form. We are looking for two binomials that will result in the given trinomial when you multiplied. Reverse FOIL Method - What you have been doing still works, but can get complicated with the leading coefficient being something other than one. try 8x2 + 10x – 3 Method 2 Example 1: Step 1: Multiply the first and last terms (6x)(-12x)=-72x2 Step 2: Find factors of -72 that will subtract or add to make +1 (coefficient of the middle term) 9x and -8x Step 3: Replace the middle term with 9x and -8x 6x2 + 9x – 8x – 12 Step 4: Factor out the Greatest Common Factor from the 1st and 2nd terms and then from the 3rd and 4th terms 6x2 + 9x – 8x – 12 3x(2x + 3) – 4(2x + 3) works like 5a – 3a = (5 – 3)(a) = 2a Step 5: Combine like terms (Final Answer) (3x – 4)(2x + 3) Step 6: Check to be sure it works … FOIL. 6x2 + 9x – 8x – 12 = Example 2: Example 3: Step 1: Step 1: Step 2: Step 2: Step 3: Step 3: Step 4: Step 4: Step 5: Step 5: Step 6: Step 6: 30 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Practice: 1) 2) 3) 4) 5) 6) HW: p.508 #8-26 even, 34 31 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 2 Factoring ax + bx + c 32 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 14 (PH Text 8.7): Factoring Special Cases Objectives: to factor perfect square trinomials and the differences of two squares A polynomial is considered completely factored when it is written as a product of prime polynomials, or one that cannot be factored. To factor a polynomial completely: 1 – Factor out the greatest monomial factor (GCF) 2 – If the polynomial has two or three terms, look for: A perfect square trinomial A difference of two squares A pair of binomial factors 3 – If there are four or more terms, group terms, if possible, in ways that can be factored. Then factor out any common polynomials. 4 – Check that each factor is prime (cannot be factored any further). 5 – Check your answer by multiplying all the factors to be sure it returns to the original polynomial. Examples: 1. 6x2 + 9x + 3 2. 20x3 – 28x2 + 8x 3. 5x4 – 50x3 + 125x2 Look for the following special cases: Factor: Difference of Two Squares x 2 64 both terms are perfect squares Perfect Square Trinomial x 2 10 x 25 st 1 & 3rd terms are perfect squares x 2 64 x 2 10 x 25 Examples: 4. 5x4 – 245x2 5. 2m3 – 36m2 + 162m 33 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Sometimes you may need to factor out the GCF before you can factor an expression into two binomials. Factor : 10 x 2 40 Practice: Factor each expression. 1) a 2 16 2) x 2 49 3) 9 x 2 25 4) 25a 2 64 5) 18 x3 32 x 6) 4h3 36h 7) 4 x 2 12 x 9 8) 25c 2 40c 16 9) 16 x 2 24 x 9 HW: p.514 #10-42 even 34 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 35 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 15 (PH Text 8.8): Factoring by Grouping Objectives: to factor higher-degree polynomials by grouping A polynomial is considered completely factored when it is written as a product of prime polynomials, or one that cannot be factored. To factor a polynomial completely: 1 – Factor out the greatest monomial factor (GCF) 2 – If the polynomial has two or three terms, look for: A perfect square trinomial A difference of two squares A pair of binomial factors 3 – If there are four or more terms, group terms, if possible, in ways that can be factored. Then factor out any common polynomials. 4 – Check that each factor is prime. 5 – Check your answer by multiplying all the factors to be sure it returns to the original polynomial. Examples: 5. 27x3 – 3xy2 6. 4m3 – 48m2 + 144m 7. 18x2 – 12x + 2 8. 8x2y3 + 4x2y2 – 12x2y 9. 2d5 – 162d 36 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Geometry Write a polynomial to express the area of each shaded region. Then write the polynomial in factored form. 37 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials In the polynomial 6(a + b) + 3(a + b), the binomial (a + b) is common to both terms. The distributive property can be used to factor out (a + b). 6(a + b) + 3(a + b) Examples: Factor. 5. 7(a + 2b) + (a + 2b) – 3(a + 2b) 6. 11(x – 3) + 7(3 – x) 7. 4d – 4 g + 9g – 9d 8. 25r – r3 – r2s + 25s 9. 49n2 – 9m2 + 24m – 16 38 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Directions for 20 and 21. A square is enclosed within another square. The area of the larger square is the given polynomial; the area of the smaller is the monomial. Write a polynomial in factored form to represent the difference of the two areas. 20. a2 + ab + b2; 9b2 21. 4c2 + 72c + 324; 25c2 Reminder: Some polynomials may contain common binomial factors. Sometimes these binomial factors are opposites, or additive inverses. The additive inverse of a is –a. Examples: Are these polynomials additive inverses of each other? 1. x – y and y – x 2. 2x + 1 and 2x – 1 3. 3t – 4 and 4 – 3t 4. 5y – 2 and 5y + 2 HW: p.519 #10-28 even, 35 39 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials Lesson 17 (PH Text 11.1): Simplifying Rational Expressions Objectives: to identify values for variables that make a rational expression undefined to simplify rational expressions Rational numbers are numbers that can be expressed as a fraction. The denominator cannot be zero. A rational expression is similar, but usually contains two polynomials. The denominator still cannot be zero. A rational expression is in its simplest form when the numerator and denominator have 1 as their only common factor. The expression will have restrictions on the variable which will prevent the denominator from being zero, called an excluded value. Step1: Factor both the numerator and the denominator. Step2: Find the restrictions on the denominator. Step3: Simplify the expression. Examples: 7 1. a 3 4. 6. Simplify and state the values for which each expression is undefined. a3 12r 2. 3. 2 9a 1 20r 2 6a 4b 18 ah ha HW: p.655 #8-32 even, 43 40 5. 3 x 2x 5x 3 7. m 2 2m 3 9 m2 2 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 41 Algebra 1 Mrs. Bondi Unit 6 Notes: Polynomials 42