POLY4 POLYNOMIALS Student Packet 4: Polynomial Arithmetic Applications
Name ___________________________ Period __________ Date ___________ POLY4 od POLYNOMIALS Student Packet 4: Polynomial Arithmetic Applications uc e STUDENT PAGES Hundred Chart Patterns 2 • Gather empirical data to form conjectures about number patterns. • Write algebraic expressions. • Practice polynomial arithmetic. • Use algebraic expressions to prove (or disprove) conjectures. • View algebra as a useful mathematical tool. 1 POLY4.2 Picture Frames • Use mathematical reasoning to create polynomial expressions that generalize patterns. • Practice polynomial arithmetic. 7 POLY4.3 Number Tricks 2 • Use algebraic expressions to generalize patterns. • Practice polynomial arithmetic • Write verbal expressions as algebraic expressions. N ot R ep r POLY4.1 pl e: D o 11 Vocabulary, Skill Builders, and Review 16 Sa m POLY4.4 Polynomials (Student Packet) POLY4 – SP Polynomial Arithmetic Application WORD BANK (POLY4) Definition or Explanation Example or Picture uc e Word od conjecture ep r deductive reasoning N pl e: D o generalization ot R empirical evidence inductive reasoning Sa m proof Polynomials (Student Packet) POLY4 – SP0 Polynomial Arithmetic Application 4.1 Hundred Chart Patterns 2 HUNDRED CHART PATTERNS 2 Set (Goals) • Go (Warmup) Example: 5 6 7 Find the “inner product.” Find the “outer product.” 6 7 = 42 5 8 = 40 ot R Pick any four consecutive numbers on the hundred chart. od • • • Gather empirical data to form conjectures about number patterns. Write algebraic expressions. Practice polynomial arithmetic. Use algebraic expressions to prove (or disprove) conjectures. View algebra as a useful mathematical tool. ep r • We will investigate patterns on the hundred chart. We will write and use algebraic expressions to prove conjectures based on the patterns. uc e Ready (Summary) 8 N 1. Compare the products: 3. 4. 21 22 23 21 22 = _____ ____ • ____ = ______ m 5. 20 pl e: D 2. o Try this for at least four other groups of four consecutive numbers. Sa 6. Write a conjecture about this pattern as a complete sentence. Polynomials (Student Packet) POLY4 – SP1 Polynomial Arithmetic Application 4.1 Hundred Chart Patterns 2 uc e HUNDRED CHART 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 21 22 23 24 25 26 27 28 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 61 62 63 64 71 72 73 ep r 29 30 39 40 47 48 49 50 56 57 58 59 60 65 66 67 68 69 70 75 76 77 78 79 80 90 ot R 38 N pl e: D 20 37 o 74 10 od 1 82 83 84 85 86 87 88 89 91 92 93 94 95 96 97 98 99 100 Sa m 81 Polynomials (Student Packet) POLY4 – SP2 Polynomial Arithmetic Application 4.1 Hundred Chart Patterns 2 PROVING A CONJECTURE uc e 1. Conjecture that the class is going to prove: od 2. Prove your conjecture by using algebra to label each of the four consecutive numbers and then by multiplying the inner and outer products. ep r ________ ________ ________ ________ Inner Product Outer Product __________________ ___________________ o N __________________ ( ________ )( ________ ) ot R ( ________ )( ________ ) Sa m pl e: D 3. Why does this prove the conjecture? Polynomials (Student Packet) POLY4 – SP3 Polynomial Arithmetic Application 4.1 Hundred Chart Patterns 2 A SECOND CONJECTURE 1. 5 Find the product of the diagonal that begins in the upper left corner Find the product of the diagonal that begins in the upper right corner od Pick any four numbers on the hundred chart that form a 2 x 2 square. uc e Now try a different experiment from the hundred chart with four numbers that form a 2 × 2 square. 6 5 • 16 = _____ 16 ep r 15 6 • _____ = _____ ot R 2. Compare them: Try this for at least three other groups of four numbers that form a 2 × 2 square. 3. 22 23 32 33 5. pl e: D o 4. ____ • ____ = ______ N 22 33 = _____ Sa m 6. What do you notice in all of these examples? Make a conjecture using a complete sentence. Polynomials (Student Packet) POLY4 – SP4 Polynomial Arithmetic Application 4.1 Hundred Chart Patterns 2 A SECOND CONJECTURE (continued) uc e 1. Conjecture that the class is going to prove: ot R ep r od 2. Prove your conjecture by using algebra to label each of the four numbers and then by multiplying each diagonal. Find the product of the diagonal that begins in the upper right corner. N Find the product of the diagonal that begins in the upper left corner. o ( ________ )( ________ ) pl e: D __________________ ( ________ )( ________ ) ___________________ ___________________ Sa m 3. Why does this prove the conjecture? Polynomials (Student Packet) POLY4 – SP5 Polynomial Arithmetic Application 4.1 Hundred Chart Patterns 2 A THIRD CONJECTURE uc e Experiment with patterns like these four numbers on a 3 x 3 square. Multiply the vertical numbers. Multiply the horizontal numbers. Do this for at least three more 3 x 3 squares. 5 16 od 14 1. ep r 25 Horizontal Product 14 16 = 224 Vertical Product 5 25 = _____ 3. 4. ot R 2. pl e: D o N 5. Write your conjecture in words. 6. Prove your conjecture algebraically: Vertical Product Sa m Horizontal Product 7. Does your conjecture hold? Polynomials (Student Packet) POLY4 – SP6 Polynomial Arithmetic Application 4.2 Picture Frames PICTURE FRAMES Set (Goals) • Go (Warmup) Write the formula for finding the area of each: a. A square with side length s. ot R 1. od Use mathematical reasoning to create polynomial expressions that generalize patterns • Practice polynomial arithmetic ep r We will create polynomial expressions that generalize a geometric pattern, and simplify the expressions. We will use our understanding of the construction of each pattern to verify the accuracy of the polynomial we created. uc e Ready (Summary) b. A rectangle with a base, b, and a height, h. For this square-inside-of-a-square picture: o 2. N c. A circle with radius r. pl e: D a. Write in words the steps you could take to find the area of the shaded region. Sa m b. Find the area of the shaded region if the side length if the larger square is 8 inches and the side length of the smaller square is 5 inches. 3. Find the area of a circle with a radius of 7 cm. Leave the answer in terms of π . Polynomials (Student Packet) POLY4 – SP7 Polynomial Arithmetic Application 4.2 Picture Frames A SQUARE PICTURE FRAME PATTERN square 9 square 10 square n 8 8 9 9 Un-simplified expression for shaded area (picture frame) 5 Simplified expression for shaded area 82 − 52 = 64 − 25 39 6 pl e: D o 10 Side length of inner square ot R Side length of outer square N Square pattern number ep r od square 8 uc e 1. Study the given square picture frame pattern, and fill in the first 3 rows of the table. 2. Sketch the picture for square n. 3. Fill in the last row of the table to find a generalized expression for the shaded area. n Sa m 4. Substitute the values from squares 8, 9, and 10 into the simplified expression for the shaded area of the n th figure as a check. Polynomials (Student Packet) POLY4 – SP8 Polynomial Arithmetic Application 4.2 Picture Frames A CIRCULAR PICTURE FRAME PATTERN circle 9 Outer circle radius Inner circle radius 7 7 3 8 8 N Shaded area (simplified) o 4 pl e: D 9 Shaded area expression (un-simplified) ot R Circle pattern number circle n od circle 8 ep r circle 7 uc e 1. Study the given circle pattern and fill in the first 3 rows of the table. Leave the answers in terms of π . 2. Sketch circle n. 3. Fill in the last row of the table to find a generalized expression for the shaded area of this circle pattern. n Sa m 4. Substitute the values from circles 7, 8, and 9 into the simplified expression for the shaded area of the n th figure as a check. Polynomials (Student Packet) POLY4 – SP9 Polynomial Arithmetic Application 4.2 Picture Frames A RECTANGULAR PICTURE FRAME PATTERN rectangle 5 rectangle 6 rectangle n 4 4×7 5 5×8 Shaded area (simplified) 2× 4 3×5 pl e: D n Shaded area expression (un-simplified) o 6 Inner rectangle dimensions ot R Outer rectangle dimensions N Rectangle pattern number ep r od rectangle 4 uc e 1. Study the given rectangle pattern and fill in the first 3 rows of the table. 2. Sketch rectangle n. 3. Fill in the last row of the table to find a generalized expression for the shaded area of this rectangle pattern. Sa m 4. Substitute the values from rectangles 4, 5, and 6 into the simplified expression for the shaded area of the n th figure as a check. Polynomials (Student Packet) POLY4 – SP10 Polynomial Arithmetic Application 4.3 Number Tricks 2 NUMBER TRICKS 2 Set (Goals) • • Use algebraic expressions to generalize patterns. Practice polynomial arithmetic Write verbal expressions as algebraic expressions. od • Go (Warmup) 1. Perform the number trick below. Directions Numbers Choose a natural number between 1 and 10. 2 Multiply your number by 2. 3 Add 8 to your answer. 4 Divide your answer by 2. Algebraic Process n 2n pl e: D o N 1 ot R Step ep r We will perform mathematical number tricks and use algebraic expressions to show how they work. uc e Ready (Summary) 5 Subtract your original number from your answer. 6 What number did you end with? Sa m 2. What is the number trick? 3. Explain why the algebraic process supports that this trick will work for all numbers? Polynomials (Student Packet) POLY4 – SP11 Polynomial Arithmetic Application 4.3 Number Tricks 2 NUMBER TRICK 1 1. Perform the number trick below. Numbers Algebraic Process n Choose a number. 2 Multiply your number by one more than the original number. 3 Add your original number. 4 Add 1. 5 Divide by 1 more than your original number n(n + 1) = ________ pl e: D o N ot R ep r 1 uc e Words od Step Subtract 1 7 What number do you have now? Sa m 6 2. What is the number trick? 3. Explain why the algebraic process supports that this trick will work for all numbers? Polynomials (Student Packet) POLY4 – SP12 Polynomial Arithmetic Application 4.3 Number Tricks 2 NUMBER TRICK 2 1. Perform the number trick below. Words Numbers Algebraic Process uc e Step n Choose a number. 2 Add 4. 3 Multiply by your original number. 4 Multiply by 4. 5 Divide by your original number. 6 Subtract 16. 7 Divide by 4. od 1 n+4 pl e: D o N ot R ep r n(n + 4) = ________ 8 What number do you have now? m 2. What is the number trick? Sa 3. Explain why the algebraic process supports that this trick will work for all numbers? Polynomials (Student Packet) POLY4 – SP13 Polynomial Arithmetic Application 4.3 Number Tricks 2 MORE NUMBER TRICKS 1. Perform the number trick below. Words Numbers Algebraic Process Choose a number. n 2 Square your number. n2 3 Add three more than four times your original number 4 Divide by 1 more than your original number. 5 Subtract the original number. 6 What is the result? Step ot R ep r n 2 + 4n + 3 Algebraic Process n o Choose a number. Numbers pl e: D 1 Words N 2. Perform the number trick below. od 1 uc e Step Square it. 3 Subtract 4. 4 Divide by 2 less than your original number. m 2 Sa 5 6 Subtract your original number What is the result? Polynomials (Student Packet) POLY4 – SP14 Polynomial Arithmetic Application 4.3 Number Tricks 2 NUMBER TRICK TEMPLATE Words Numbers Algebraic Process Sa m pl e: D o N ot R ep r od Step uc e Use this page to create your own number trick. Polynomials (Student Packet) POLY4 – SP15 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (POLY 4) od Choose a number. Add 4. Multiply by 2. Subtract 8. Divide by your original number. The result is… ep r • • • • • • uc e Here is a number trick. Felicity did some work to verify this trick. Match her work with the word or words that describe what she did. You may use a word more than once. First she tried it with numbers. ot R 1. _____ A. conjecture 5 9 18 10 2 B. deductive reasoning 12 16 32 24 2 “I think the result is always going to be 2.” Then she drew these pictures. pl e: D 3._____ N _____ Then she said, o 2._____ C. empirical evidence D. generalization E. inductive reasoning _____ _____ m 4._____ Then she said, “This shows that the result is always going to be 2.” Sa _____ F. Proof Polynomials (Student Packet) POLY4 – SP16 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 1 1. ________________ 25 x 2 − 15 x = 2. uc e Use an area model to factor. First find the GCF of the terms. 4 x 2 + 12 x − 20 ______________ ep r od GCF ot R When factoring a binomial: • First look for the GCF of all the terms • Then look for the “difference of two squares” pattern N Factor completely. If it cannot be factored, write “not factorable.” 3. 2x2 + 8 4. 4 x 2 − 16 x 5. 12 x 2 − 7 pl e: D o When factoring a trinomial: • First look for the GCF of all the terms • Then look for a quadratic trinomial where coefficient of the square term is 1 • Then look for a quadratic trinomial where the coefficient of the square term is not 1 Factor completely. If it cannot be factored, write “not factorable.” 7. 3 x 2 + 12 x − 9 8. 4 x 2 + 7 x − 5 Sa m 6. 5 x 2 + 15 x − 20 Polynomials (Student Packet) POLY4 – SP17 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review Rewrite each expression as a power of 2. 1. 2. 2 (42) (42) 43 3. 4. 6. ( ) ep r 5. 4 •2 24 • 45 7 7 2 4 3 22 (82 )3 N ot R 8 83 1 16 od -8 uc e SKILL BUILDER 2 9. pl e: D o Evaluate when a = 1 , b = -2, c = -2 8. 7. -b b 2 – 4ac 2a 10. 11. -b − b 2 − 4ac 2a Sa m -b + b 2 − 4ac 2a b 2 – 4ac 2a Polynomials (Student Packet) POLY4 – SP18 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 y -6 1 -3 2 0 3 3 4 6 ep r od x 0 uc e Find equations of lines in different forms. Use the information given. 1. Given: (graph) 2. Given (table): slope-intercept form ot R slope-intercept form point-slope form point-slope form standard form o N standard form Evaluate each expression if x = - 3 pl e: D 2 3. x4 4. x3 5. x2 6. x1 7. x0 8. x-1 9. x-2 10. x-3 m 11. Examine your answers to problems 5-12. Under what conditions is the result Sa positive? negative? a fraction between 0 and 1? Polynomials (Student Packet) POLY4 – SP19 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review Simplify each expression. 1. 2. 44 3. 9 121 5. 6. ot R 4. 120 48 75 ep r − 3 12 13 49 N 9 8. (2 + 2 ) pl e: D 4( 2 + 2 ) o 7. 10. 11. 9. (2 + 2 )(2 − 2 ) 2 3 24 6 −5 + 52 − 4(2)(3) 2(2) Sa m 16 − 421 8 12. 2 40 16 + 9 5 od A radical expression containing square roots is simplified when there are: • no perfect squares under the radical • no fractions under the radical • no radicals in the denominator. uc e SKILL BUILDER 4 Polynomials (Student Packet) POLY4 – SP20 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 5 Write each polynomial as a sum of terms in decreasing order of powers. 3. x – 2(x + 3)(x – 2) uc e 2. 6 – (x – 2)(x – 3) ep r od 1. 5 + (x + 3)(x – 3 ) ot R Factor completely. If it cannot be factored, write “not factorable.” 4. x2 – 7x + 10 5. x2 + 10x + 25 6. x2 - 64 8. x2 + 15x + 26 9. 2x2 + 8x + 3 pl e: D o N 7. -50x + 30 Use > , < , or = to make each statement true. Show work. −1 −4 10. 11. _____ 4 73 • 7 −3 _____ 7 7 −3 1 2 1 2 (9 • 16) _____ 9 16 m 4 12. 3 Sa Determine which numbers are in scientific notation. If NOT, write it in scientific notation. 13. 8.21 x 105 Polynomials (Student Packet) 14. 0.213 x 10-4 POLY4 – SP21 1 2 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Simplify each radical expression. 2. 3. 59 ? _____ and _____ 4. 6 4 49 14 )( 25 −2 − 25 ) 6. 27 7. − 49 8. 32 50 9. 4 10. 17 81 N 3 12 2 18 ot R 5. ep r od 6− 4 ( −2 + uc e 1. Between which two consecutive integers is pl e: D o 36 30 − 32 24 12. -3 ≥ 33x + 8 Sa m Solve and graph: 11. -2 x + 7 ≤ 5 Polynomials (Student Packet) POLY4 – SP22 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 7 1 6 x − 1 + x = -36 3 0.5 x + 4 = -2 x − 6 2. ot R ep r od 1. uc e Solve for x. Check by substitution. 4. Write the equation in slope-intercept form. N Given graph: 6. State the x- and y- intercepts. pl e: D o 5. Write the equation in standard form. Compute: (-3)2 m 7. -12(-6 + 4) 5 – 32 9. -32 11. 36 – 12 (-6 + 4) 12. 36 ÷ 12(-6 + 4) Sa 10. 8. Polynomials (Student Packet) POLY4 – SP23 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 8 od uc e Simplify each expression by combining like terms. Write each polynomial as a sum. 1. 2 x − 5 y + 6 x − 8 y 2. -13 − x + 5 y − 7 − 2 x ( xy − 4 x 2 ) − (3 x 2 − 5 xy ) 4. 6 x − 9 − 2( x − 7) 5. -5( x − y ) − 3( y − x ) 6. x(2 x + y ) + y (2 x + y ) 7. ( x − 6)( x + 6) N ot R ep r 3. ( x − 7)( x − 10) pl e: D o 8. ( x − 3)(- x + 8) 10. (5 − x )( x + 9) m 8. Sa 11. Write a variable expression for the area of a square whose side is x + 8. Polynomials (Student Packet) POLY4 – SP24 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 9 x is less than -2 2. x is greater or equal to than -2 3. the opposite of x is less than or equal to -3 4. 3 is less than x 5. -2 is less than or equal to the opposite of x. ot R ep r od 1. uc e Write each statement using symbols. If the variable is on the right side, change it to the left side using appropriate properties. Then graph each. Sa m pl e: D o N Graph each inequality. Be sure they are in slope-intercept form first. 6. 7. 2 y > x +1 -2y + x ≥ 10 3 8. Describe the differences between the graph of an inequality in one variable and the graph of an inequality in two variables. Polynomials (Student Packet) POLY4 – SP25 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 10 2. 15 x 2 − 35 x 3. 4. x 2 + 10 x + 9 5. x 2 − 4 x − 77 6. 7. x 2 − 15 x + 26 8. x 2 − 30 x + 200 10. 10 x 2 + 80 x + 160 11. 7 y − 49 y 2 od 8 x 2 + 40 x x 2 + 9 x − 36 9. x 2 + 2x + 6 2 x 2 − 4 x − 30 12. 3 x 2 + 18 x − 3 15. x 2 − 10 18. x 2 − 24 x − 144 pl e: D o N ot R ep r 1. uc e Factor using any method. Be sure to factor each polynomial completely. If it cannot be factored write “not factorable.” x2 − 9 14. 2 x 2 − 50 16. −3 x 2 − 18 x 17. x 2 + 24 x + 144 Sa m 13. Polynomials (Student Packet) POLY4 – SP26 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 11 Graph each system of inequalities. Test several points to verify correct shading. y ≤ -3x + 5 2. 1 y – 3 < -2 x 5 uc e 1. 2x + y ≥ -4 y > - x pl e: D Complete the table. o N ot R ep r od 3 Fraction Ex. 0.009 9 1,000 1. 0.0028 m Decimal Sa 2. 3. 4. Polynomials (Student Packet) Product of a number between 1 and 10, and a multiple of 10 9 × 0.001 or 9 × 1 103 Scientific notation 9 × 10-3 4.76 100 3.5 × 1 107 4.2 × 10 −3 POLY4 – SP27 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review SKILL BUILDER – POLY 12 6x – 2y = -16 4x + y = 1 2. 4x = 6 + y 1 2x – y = 3 2 ep r od 1. uc e Solve each system using algebra. o N ot R Arnon wants to record the 12-hour opera marathon on the radio. He has 90-minute discs and 60-minute discs. If he uses 9 discs, how many of each type will he use? 3. Solve the algebraically problem using 4. Solve the problem algebraically using one variable. two variables (a system of equations). pl e: D 5. Could you solve this problem without algebra? Explain. Sa m 6. Think of a number. Subtract 7. Multiply by 3. Add 30. Divide by 3. Subtract the original number. The result is always 3. Use polynomials to illustrate this number trick. Polynomials (Student Packet) POLY4 – SP28 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review TEST PREPARATION (POLY4) uc e Show your work on a separate sheet of paper and choose the best answer. n + 10 + 4 B. n+5 C. n + 14 D. n + 54 ep r A. od 1. Below is an excerpt from a hundreds chart. If 40 = n , which expression(s) represent(s) 54? 40 41 42 43 44 45 50 51 52 53 54 55 60 61 62 63 64 65 A. 2n + 1 C. n2 + 1 ot R 2. Below is a 3 × 3 square taken from a hundreds chart. What expression (in terms of n) should go in the square with the question mark? n B. 3n + 1 D. n + 21 ? pl e: D o N 3. Below is a 2 × 2 square taken from a hundreds chart. What expression (in terms of n) represents ab. a b n (n + 1)(n + 10) (n − 1)(n − 10) A. B. C. (n + 1)(n − 10) D. (n − 1)(n + 10) 4. If n is a number, what is an expression for four more than four times a number. 4n + 4 B. n(4 + 4) C. (4 + 4) × n D. 4+4+n m A. Sa 5. The radius of the smaller circle is given. If the radius of the larger circle is twice the radius of the smaller circle, what is the area of the shaded part? A. 8π B. 36π C. 64π D. 48π Polynomials (Student Packet) 4 POLY4 – SP29 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (POLY4) uc e 4.1 Hundred Chart Proofs 2 1. Below is an excerpt from a hundreds chart. If 52 = n , what does 62 represent? Give the most simplified answer. ep r 2. 1 11 4 14 1× 14 = 14 36 46 39 49 36 × 49 = 1,764 46 × 39 = 1,794 74 84 ot R 11× 14 = 44 od 40 41 42 43 44 45 50 51 52 53 54 55 60 61 62 63 64 65 77 87 74 × 87 = 6,438 84 × 77 = 6,468 N If the following table follows the pattern above, write a conjecture for the relationship between ad and bc. b d pl e: D o a c 4.2 Picture Frames Use the following picture to answer questions 3 and 4. 3. Write an expression for the area of the smaller rectangle. 4 n n+3 7 Sa m 4. Write an expression for the area of the shaded part. 4.3 Proving Number Tricks 2 5. If n is a number, write an expression for “the product of a number and one more than the number.” Polynomials (Student Packet) POLY4 – SP30 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review Sa m pl e: D o N ot R ep r od uc e This page is left blank intentionally. Polynomials (Student Packet) POLY4 – SP31 Polynomial Arithmetic Application 4.4 Vocabulary, Skill Builders, and Review Sa m pl e: D o N ot R ep r od uc e This page is left blank intentionally. Polynomials (Student Packet) POLY4 – SP32 Polynomial Arithmetic Application uc e HOME-SCHOOL CONNECTION (POLY4) For problems 1 and 2, consider the 3 × 3 square below taken from a hundreds chart. a b od n ot R 2. Multiply ab so that it is a sum of terms ep r 1. What expression (in terms of n) should go in the square with the a and what expression (in terms of n) should go in the square with the b? Sa m pl e: D o N 3. If the length of the smaller square is n and the length of the larger square is 12, what is an expression for the area of the shaded part? Signature ___________________________________ Polynomials (Student Packet) Date________________ POLY4 – SP33 Polynomial Arithmetic Application COMMON CORE STATE STANDARDS – MATHEMATICS STANDARDS FOR MATHEMATICAL CONTENT Interpret expressions that represent a quantity in terms of its context: Interpret parts of an expression, such as terms, factors, and coefficients. A-SSE-1b Interpret expressions that represent a quantity in terms of its context: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret n P(1+r) as the product of P and a factor not depending on P. A-SSE-2 Use the structure of an expression to identify ways to rewrite it. For example, see x – y as 2 2 2 2 (x ) – (y ) , thus recognizing it as a difference of squares that can be factored as 2 2 2 2 (x – y )(x + y ). CA Addition (CA.A.8a) Use the distributive property to express a sum of terms with a common factor as a multiple of a 2 2 sum of terms with no common factor. For example, express xy + x y as xy (y + x). CA Addition (CA.A.8b) Use the properties of operations to express a product of a sum of terms as a sum of products. For example, use the properties of operations to express (x + 5)(3 - x + c) as 2 -x + cx - 2x + 5c + 15. A-APR-1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials, and divide polynomials by monomials. Solve problems in and out of context. A-APR-4 Prove polynomial identities and use them to describe numerical relationships. For example, the 2 2 2 2 2 2 2 polynomial identity (x + y ) = (x – y ) + (2xy) can be used to generate Pythagorean triples. uc e A-SSE-1a 4 N ot R ep r od 4 STANDARDS FOR MATHEMATICAL PRACTICE o Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. pl e: D MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8 DO NOT DUPLICATE © 2012 Sa m First Printing Polynomials (Student Packet) POLY4 – SP34
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