Introduction to Polynomials Math Background Previously, you Identified the components in an algebraic epression Factored quadratic epressions using special patterns, grouping method and the ac method Worked with properties of integers Identified the zeros or roots of a quadratic function Recognized special patterns perfect squares and the difference of squares In this unit you will Identify polynomials and perform operations with polynomials Use the Binomial Theorem to epand binomials raised to positive integer powers Factor and divide polynomials Use the Factor Theorem and Remainder Theorem to factor polynomials completely You can use the skills in this unit to Analyze the effect of a coefficient and a constant on a variable. Identify sums, differences, and products of polynomials as polynomials. Use polynomial identities to describe numerical relationships. To prove polynomial identities using properties of operations. Factor sum and difference of two cubes. Apply Pascal s Triangle to the epansion of binomials. Use the Binomial Theorem to calculate a specific term in a polynomial or to epand binomials. Factor a polynomial using synthetic or long division. Vocabulary Algebraic Epression It is a combination of numbers, variables and operation signs to represent a certain quantity. Binomial A polynomial representing the addition or subtraction of eactly two distinct terms. Binomial Theorem A rule for writing out the epansion of a binomial raised to any positive integer power. Closed Describing a set for which a given operation (such as addition or multiplication) gives a result that is also a member of the same set. Combinations The number of possible ways of selecting m objects out of n objects when you don t care about the order in which the m objects are arranged. Factor Theorem It states that if f( a) 0in which f ( ) represents a polynomial in, then ( a) is one of the factors of f ( ). Monomial An algebraic epression consisting of only one term. It is either a constant, a variable, or a product of a constant and one or more variables. Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 1 of 0 9/10/014
Pascal s Triangle A triangular array of numbers in which each row starts and ends with 1 and each number in between is the sum of the pair of numbers above it. The number at the ape is 1. The numbers at the nth row of Pascal s triangle are the same as the coefficients of and y in the epansion of 1 ( y) n. Polynomial An algebraic epression that consists of two or more terms. Polynomial Identity They are patterns involving sums and differences of like powers. Remainder Theorem The theorem stating that if a polynomial in, f ( ), is divided by ( a), where a is any real or comple number, then the remainder is f ( a ). Root The value(s) of a variable that makes the equation true. Synthetic Division A method of performing polynomial long division, with less writing and fewer calculations. Synthetic Substitution The process of using synthetic division to evaluate f ( c) for a polynomial f ( ) and a number c. Essential Questions Just as operations performed in the subset of decimals or set of integers yield decimals and integers, respectively, do the operations of addition, subtraction, and multiplication across polynomials yield polynomials? Why do we rewrite polynomials epressions? Overall Big Ideas The system of polynomials is closed under the operations of addition, subtraction, and multiplication. Polynomial identities allow us to derive and generate numerical and polynomial relationships such as those seen in Pythagorean triples and the binomial theorem. Polynomial epressions can sometimes be rewritten to find relevant characteristics for the situation including finding zeros or etrema. Skill To identify, classify, evaluate, add and subtract polynomials. To multiply polynomials and use binomial epansions to epand a binomial epression raised to positive integer powers. To factor polynomials. To divide polynomials. To apply the Factor Theorem and Remainder Theorem. Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page of 0 9/10/014
Related Standards A.SSE.A.1a Interpret parts of an epression, such as terms, factors, and coefficients. *(Modeling Standard) A.APR.A.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 of degree and higher. A.APR.C.4 Prove polynomial identities and use them to describe numerical relationships. For eample, the polynomial identity y y y can be used to generate Pythagorean triples. A.APR.C.5 Know and apply the Binomial Theorem for the epansion of y n in powers of and y for a positive integer n, where and y are any numbers, with coefficients determined for eample by Pascal s Triangle. A.APR.B. Know and apply the Remainder Theorem: For a polynomial p( ) and a number a, the remainder on division by a is p( a ), so pa ( ) 0if and only if ( a) is a factor of p( ). A.APR.B. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A.SSE.A.- Use the structure of an epression, including polynomial and rational epressions, to identify ways 4 4 to rewrite it. For eample, see y y, thus recognizing it as a difference of squares as that can be factored as y y. A.APR.D.6 a ( ) Rewrite simple rational epressions in different forms; write b ( ) in the form r ( ) q ( ), where b ( ) a ( ), b ( ), q ( ), and rare ( ) polynomials with the degree of rless ( ) than the degree of b ( ), using inspection, long division, or, for the more complicated eamples, a computer algebra system. Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page of 0 9/10/014
Notes, Eamples, and Eam Questions UNIT 4.1 Identify, classify, evaluate, add and subtract polynomials f a a... a a a, a 0 n n1 Polynomial Function: a function of the form n n1 1 0 Note: Eponents are whole numbers and coefficients are real numbers. n Leading Coefficient: a n Constant Term: a 0 Degree: n, the largest eponent of Standard Form of a Polynomial Function: The terms are written in descending order of the eponents Classifying Polynomial Functions: I. Number of Terms 1 term = monomial terms = binomial terms = trinomial 4 or more terms = polynomial This is kind of tricky but a n is the name of the coefficient with the same degree. So, a n is the coefficient of the term that is the n th degree and a n 1 is the coefficient of the term that is degree n 1. II. Degree Degree Name (Degree) Standard Form Eample Classification of Eample y Constant Monomial 0 Constant f a 1 Linear f a b Quadratic f a b c Cubic f a b c d 4 Quartic 4 f a b c de y Linear Monomial y 5 7 Quadratic Trinomial y 5 Cubic Binomial 4 y 9 1 Quartic Polynomial Identifying Polynomial Functions f 5 8 a polynomial function? If yes, write it in standard form. E 1: Is No. In order to be a polynomial function, all eponents must be whole numbers. 5 f 8 a polynomial function? If yes, write it in standard form. E : Is 4 Yes. All eponents are whole numbers and all coefficients are real numbers. f 8 Note: This is a quartic trinomial (degree = 4). Standard Form: 4 5 Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 4 of 0 9/10/014
Evaluating Polynomial Functions Using Direct Substitution 4 E : Find f if Adding Polynomials f 5 6 1. 4 f 5 6 1 16 85 4 11 48 8 0 1 1 40 0 1 1 60 1 1 48 1 47 E 4: Add the polynomials 5 4 10 1 17 4 Subtracting Polynomials. So : f () 47 Vertical Method: Write each polynomial in standard form and line up like terms. Then, add the like terms. Note the second polynomial does not have a linear term so 0 is added to the line. 4 5 101 4 17 0 1 4 10 4 E 5: Subtract the polynomials 9 4 8 7 7. To subtract, we will rewrite the problem as an addition problem by adding the opposite. 9 4 8 7 7 9 4 87 7 Horizontal Method: Combine each set of like terms. 1111 Write the final answer in standard form. 11 11 E 6: Subtract the polynomials 5 4 6 1 5 8 7. 5 4 6 15 8 7 Horizontal Method: 4 5 6 1 5 87 4 5 11 10 5 Vertical Method: 4 5 6 1 5 4 5 11 10 Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 5 of 0 9/10/014 8 7 5
SAMPLE EXAM QUESTIONS 1. Which epression represents the perimeter of the rectangle? + 4 + 5 (A) 6 + 8 (C) 6 + 16 (B) 1 + 8 (D) 1 + 16 Ans: D. Which epression represents the perimeter of the triangle shown below? + 4 + 1 (A) (B) + 4 5 (C) 5 6 (D) 8 Ans: D. The function g is the amount of money Shawn has in the bank at the beginning of the month. The function f is the amount of money withdrawn from the account during the month. Which epression represents the amount of money left at the end of the month? f( ) 1 g ( ) 6 0 (A) (B) 5 5 8 (C) 5 8 (D) 5 8 5 5 8 Ans: C Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 6 of 0 9/10/014
4. Subtract the following polynomials: (A) 4y 7y5y 5y y y (C) y 1y8 (B) 6 y y (D) 6y 1y 8 Ans: C UNIT 4. Multiply Polynomials and Binomial Epansion Multiplying Polynomials E 7: Find the product 4. Horizontal Method: Use the distributive property by distributing each term of the first polynomial. 4 4 4 4 4 6 8 5 4 5 4 4 4 6 8 Combine like terms and write the answer in standard form. 5 4 5 10 8 E 8: Multiply the polynomials 1 4 7. Vertical Method: Use long multiplication. 4 7 1 4 6 1 1 4 7 4 6 1 5 7 E 9: Multiply the polynomials 5 14. Multiply the polynomials two at a time. Because they are binomials, we can use FOIL to multiply the first two. 1054 954 Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 7 of 0 9/10/014
954 95 Use the distributive property. 8 60 9 5 Combine like terms and write in standard form. 41 0 Review: Special Products (Allow students to come up with these on their own.) Memorize these! Sum and Difference Product a bab a b Square of a Binomial Cube of a Binomial a b a abb a b a abb a b a a bab b a b a a bab b E 10: Simplify the epression y. Using the cube of a binomial: y y y 7y 54y 6y 8 Application Problems E 11: Find a polynomial epression for the volume of a rectangular prism with sides, 4, and. Volume of a Rectangular Prism = Length Width Height FOIL: 4 1 Vertical Method: 1 1 4 1 4 4 E 1: From 1985 through 1996, the number of flu shots given in one city can be modeled by A t t t t 4 11. 8.5 194 4190 759 for adults and by C t t t t Write a model for the total number F of flu shots given in these years. 4 6.87 106 51 15 540 for children, where t is the number of years since 1985. Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 8 of 0 9/10/014
To find the total flu shots, we need to add the polynomials. Vertical Method: Solution: 4 11. 8.5 194 4190 759 t t t t 4 6.87 106 51 15 540 t t t t 4 18. 97.675 194 4055 81 t t t t F t t t t 4 18. 97.675 194 4055 81 QOD: What is the advantage of the vertical method when adding, subtracting, or multiplying polynomials? BINOMIAL EXPANSION: Eploration: Epand the following. Notice the pattern???? 0 1 1 1 1 1 1 1 1 1 1 1 1 Pascal s Triangle 1 Row 0 1 1 Row 1 1 1 Row 1 1 Row 1 4 6 4 1 Row 4 1 5 10 10 5 1 Row 5 This pattern can be continued The values in the triangle correspond to the coefficients of the binomial epansion of 1 n. They also correspond to n C r : 1 0 0! 1 C 1 0! 0 0! 1 0 0 1! 1 C 1 0! 1 0! 1 1 1 1! 1 C 1 1! 1 1! 1! C 1 0! 0! 0 C!! C 1!! 1 1! 1! Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 9 of 0 9/10/014
Notation: Combination (Binomial Coefficient) n C r n n! r r! n r! Epanding a Binomial The Binomial Theorem E 1: Epand 4. Since n = 4, we use row 4 from Pascal s Triangle for the coefficients: 1 4 6 4 1 Eponent of first term () starts at 4 and goes down by 1 each time; eponent of second term ( ) starts at 0 and goes up by 1 each time. 1 4 4 0 1 1 0 4 4 1 4 6 4 1 54 10881 NOTE: When epanding a binomial: The eponent of the first term starts at n and goes down by 1 each time. The eponent of the second term starts at 0 and goes up by 1 each time. The sum of the two eponents of each term should equal n. If the second term is negative, the terms of the epansion alternate signs, like in the eample above. We can also use the binomial theorem to find the epansion. Using the formula, we get: 4 4 4 4 4 0 1 4 4 4 1 4 ( ) 6 ( 9) 4 ( 7) 1(81) 1 54 10881 4 40 0 41 1 4 4 44 4 ( ) ( ) ( ) ( ) ( ) c d. E 14: Epand 5 Since n = 5, we use row 5 from Pascal s Triangle for the coefficients: 1 5 10 10 5 1 Eponent of first term (c) starts at 5 and goes down by 1 each time; eponent of second term (d) starts at 0 and goes up by 1 each time. cd 1c d 5c d 10c d 10c d 5c d 1c d 5 5 0 4 1 1 4 0 5 5 4 4 5 4c 1 581c d 107c 4d 109c 4d 5c16d 1 d 5 4 4 5 4c 810c d 1080c d 60c d 40cd d **A common mistake made by students is that the variable of each term is raised to appropriate powers but not the coefficient. For instance, one may think that the first term in the epansion in the eample above is c 5 instead of (c) 5 = 4c 5. Also, students may forget to use the numbers from Pascal s Triangle in an epansion or will not multiply all coefficients together and simplify for each term. Finding a Particular Term of a Binomial Epansion: Recall that the coefficients from Pascal s Triangle can be found individually by evaluating n C r n n! r, or by using technology. r! n r! Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 10 of 0 9/10/014
E 15: Find the fifth term of the binomial epansion of 1 10. The first term of the epansion has r 0, so the fifth term has r 4 in the combination formula. 10 10! 10 9 87 6 1 64 64 1440 4 4!104! 4 Fifth term: 6 4 6 6 E 16: Find the coefficient of the term with 7 a in the epansion of 4a b 11. Sum of eponents = 11, so the term has b 4 ; therefore, r 4. 11 7 4 4 a b 0 1684 a 81 b 47,944, 0ab 4 7 4 7 4 11 To evaluate on the TI-84, use the following steps: Type 11 into the calculator on the home screen, press 4 MATH, arrow over to PRB, find n C r,hit enter and then type in 4. QOD: Is the sum of EVERY row of Pascal s triangle an even number? Eplain. SAMPLE EXAM QUESTIONS 1. Which polynomial represents the product of 8? A. B. C. D. 8 16 4 4 4 4 8 16 16 16 Ans: A. Suppose y 9 and ( y) 1. What is y? A. B. 1 C. 6 D. 81 Ans: A Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 11 of 0 9/10/014
. What is the product of the polynomials? 5 4 6 A. B. C. D. 15 10 1 15 6 1 15 10 1 15 6 1 Ans: C 4. What is the 4 th term of the epanded binomial ( 1)? 6 A. C. 40 B. 40 D. 60 160 Ans: D UNIT 4. Factor Polynomials Review: Factoring Patterns Factoring a General Trinomial E 17: Factor the trinomial. 5 1 ac Method: ac 4 Split the middle term: Factor by grouping: 8 4 8 5 8 1 4 4 4 4 Factoring a Perfect Square Trinomial E 18: Factor the trinomial 6 9. Difference of Two Squares Use a abb a b. E 19: Factor 6 49 y. Use a b aba b. 6 7y 6 7y6 7y Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 1 of 0 9/10/014
Common Monomial Factor E 0: Factor the trinomial completely. 7 Factor the GCF and the binomial square. 4 16 4 4. Since this is not completely factored, use a b a ba b 4 Sum and Difference of Two Cubes a b ab a abb a b ab a abb E 1: Factor the binomial E : Factor the binomial 8 1. Use a b a ba ab b. y 7. 6 Use a b a ba ab b. 1 1 11 14 1 y y y y 4 y y y 9 Factoring by Grouping E : Factor the polynomial 9 18. Group each pair of terms and factor the GCF. 9 Factor the common binomial factor. 9 Factor the remaining terms if possible. Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 1 of 0 9/10/014
6 4 E 4: Factor the polynomial 4 0 4. Factor out the common binomial factor. 4 4 5 6 Factor the remaining trinomial. 4 QOD: Give an eample of a binomial that can be factored either as the difference of two squares or as the difference of two cubes. Show the complete factorization of your binomial. SAMPLE EXAM QUESTIONS 1. What is the factored form of the polynomial 7? A. 9 B. 9 C. 9 D. 9 Ans: D. Which epression is equivalent to c b yc yb? A. by c B. cy b C. yb c Ans: C 4. Which is equivalent to 4 9y A. y B. y y C. y y y Ans: A Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 14 of 0 9/10/014
UNIT 4.4 Dividing Polynomials Review: Long Division 4,581 1894 4581 05 184 18 07 111 9 19 Remainder is 19, so the quotient is 19 1894 Long Division of Polynomials (same process!) The closure property states that when you combine any two elements of the set, the result is also in that set. Polynomials are closed with respect to addition, subtraction and multiplication. When we add, subtract or multiply polynomials, our result is a polynomial. However, when we divide polynomials, we might not get a polynomial. Therefore, we say that division of polynomials is not closed. E 5: Find the quotient. 4 8 116 Note: Every term of the polynomial in the dividend must be represented. Since this polynomial is missing an term, we must include the term 0. 11 88 4 8 0 116 4 We place the remainder, 58, over the divisor. Just like long division with natural 11 0 numbers. 11 11 99 88 6 58 Solution: 11 88 88 64 58 Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 15 of 0 9/10/014
E 6: Find the quotient of 4 y y y 5 and y y 1. 4 y y y y y y 1 0 5 y y y 4 y + y y y y y y y (add the opposite) y y 5 (bring down the net term) y y (remainder) Remember to put a place for the missing term. Solution: y y y y1 Dividing Polynomials Using Synthetic Division: (Note: This procedure can only be used when the divisor is in the form k (a linear binomial).) E 7: Divide the polynomial 7 6 by. With the polynomial in standard form, write the coefficients in a row. If a term is missing, make sure to put a zero in the row. Put the k value (-) to the upper left in the bo. Bring down the first coefficient, then multiply by the k value. Add straight down the columns and repeat. 1 7 6 18 1 6 11 R 7 Note for graphing: This means that (, 7) is an ordered pair that is on the graph of the function. The coefficients of the quotient and remainder appear in synthetic substitution. Quotient: 7 1 611 **Notice the quotient is now one degree lower than the dividend (the original polynomial). Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 16 of 0 9/10/014
Synthetic Substitution: 4 E 8: Find f if f 5 6 1 using synthetic substitution. Using the polynomial in standard form, write the coefficients in a row. Put the -value to the upper left. 1 5 6 1 Bring down the first coefficient, then multiply by the -value. 1 5 6 1 multiply 6 Add straight down the columns, and repeat. 1 5 6 1 6 10 0 48 5 15 4 47 The number in the bottom right is the value of f. Eploration: Use the polynomial function f 5. Use long division to divide Then use synthetic substitution to find f. What do you notice? SAMPLE EXAM QUESTIONS So : f () 47 f by. 1. Divide 4 ( 7) ( 1) using long division. (A) 1 6 1 (B) 6 1 1 (C) 6 1 1 (D) 6 1 1 Ans: D. The volume V( ) and height ( h ) of the prism is given. Find a polynomial epression for the area of the base ( B ) in terms of. (Hint: V Bh) (A) (B) (C) 6 (D) 4 4 5 h Ans: A V( ) 5 4 Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 17 of 0 9/10/014
. Write an epression that represents the width of a rectangle with length 5 and area 1 47 60. (A) (B) (C) (D) 7 1 7 1 50 17 8 5 70 5 17 1 Ans: B Unit 4.5 Factor Theorem and Remainder Theorem Remainder Theorem: If a polynomial f is divided by k, then the remainder is r f k. E 9: Find the remainder without using division for 4 8 11 6 divided by. 4 4 f 8 11 6 r f 8 11 6 8 5 Note: Compare your answer to the long division problem E 5 above. 5 E 0: Find f if f 7 11 using synthetic substitution. This polynomial function is in standard form, however it is missing two terms. We can rewrite the 5 4 function as f 0 7 0 11 to fill in the missing terms. 1 0 7 0 11 9 1 4 16 1 7 14 4 17 **With the remainder theorem, we can find this value much quicker. 5 r f 7 11 17 This also means that (, 17) is an ordered pair that would be a point on the graph. f 17 Factor Theorem: A polynomial f has a factor k if and only if f k 0. E 1: Determine if is a factor of 5 1 without dividing. Show that r f 0 51 0, so by the Factor Theorem, : is a factor of. 5 1 Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 18 of 0 9/10/014
E : Factor f 14 8 4 given that f 6 0. Because f 6 0, we know that 6 is a factor of f synthetic division to find the other factors. 6 14 8 4 18 4 4 4 4 0 f f 6 44 6 by the Factor Theorem. We will use Note for graphing: This means that ( 6,0) is an ordered pair that is on the graph of the function. 6 is called a zero. It is also an intercept. E : One zero of f 9 1 is 7. Completely factor the function. f 7 0, we know that 7 Because synthetic division to find the other factors. 7 9 1 14 5 1 5 0 is a factor of f f 7 5 f 7 1 by the Factor Theorem. We will use QOD: If f is a polynomial that has a as a factor, what do you know about the value of f a? SAMPLE EXAM QUESTIONS 1. Use the remainder theorem to find the remainder when f() is divided by - c? f 4 ( ) 8 1 ; 1 A. 1 B. -1 C. 5 D. -5 Ans: C. Use the factor theorem to determine whether c is a factor of f(). f 4 ( ) 5 19 4 4; 4 A. Yes B. No Ans: A Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 19 of 0 9/10/014
. Use the remainder theorem to find the remainder when f() is divided by -k. f k ( ) 1; A. 15 B. 9 C. 18 D. Ans: D 4. Use the factor theorem to determine whether the first polynomial is a factor of the second polynomial. f 1; ( ) 1 A. No solution B. no C. yes D. 0 Ans: C Alg II Notes Unit 4.1 4.5 Introduction to Polynomials Page 0 of 0 9/10/014