11-1 11-2 11-3 11-4 11-5 11-6 11-7 11-8 (dropped) Chapter Review Algebra Chapter 11 11-1 11-2 11-3 11-4 11-5 11-6 11-7 11-8 (dropped) Chapter Review

11-1 Investments and Polynomials 11.1 11-1 Investments and Polynomials Objectives: Add and subtract polynomials Translate investment situations into polynomials When amounts are invested periodically and earn interest from the time of investment, the total value can be represented by a polynomial.

Scale Factor: the amount of increase to be multiplied - interest rate Definitions: Polynomial in x – is a sum of multiples of powers of x. For example: -2x3 + 3x2 + 2x + 6 Standard form for polynomials are polynomials written in decreasing powers of x. Scale Factor: the amount of increase to be multiplied - interest rate 11.1

11.1 1) As a New Year's resolution, Bert has decided to deposit $100 in a savings account every January 2nd. The account yields 3% interest annually. How much will his savings be worth when he makes his fourth deposit? Jan 2nd – 1st deposit = $100 Jan 2nd – 2nd deposit 100(1.03) + 100 = $203 Jan 2nd – 3rd deposit 100(1.03)2 + 100(1.03) +100 = $309.09 Jan 2nd – 4th deposit 100(1.03)3 + 100(1.03)2 + 100(1.03) +100 = $418.36

11.1 2) Janice has a savings account that has a scale factor of x. She makes deposits at regular yearly intervals. The first year she deposits $800, the second year $300, the third year $450, and the fourth year $775. What is her balance immediately after the fourth deposit? Jan 2nd – 1st deposit = 800 Jan 2nd – 2nd deposit = 800x + 300 Jan 2nd – 3rd deposit = 800x2 + 300x + 450 Jan 2nd – 4th deposit = 800x3 + 300x2 + 450x + 775

11.1 3) Which is more advantageous, to invest $50 per year for four years or to invest $100 in the first year and $100 in the fourth year? In both instances, the money earns 3% interest a year. $50 per year – 4th deposit = 50(1.03)3 + 50(1.03)2 + 50(1.03) + 50 = $209.18 $100 1st & 4th – 4th deposit = 100(1.03)3 + 100 = $209.27 You’ll make $0.09 more depositing $100 in the 1st and 4th years

4) Simplify a) 4x4 + 2x3 - x - 4x(x2 - 4x + 2) b) 5x4 + 2x2 - x + (?) = x2 - 4x + 2 11.1 4x4 + 2x3 - x - 4x3 + 16x2 - 8x Distribute -4x first 4x4 + 2x3 - 4x3 + 16x2 - x - 8x Use Commutative Property – get like terms together. 4x4 - 2x3 + 16x2 - 9x Simplify & standard form - 5x4 -5x4 Get rid of the x4 term - 2x2 = (-5x4) +x2 -2x2 - 4x + 2 Decrease x2 term by 2x2 + x = (-5x4-x2) -4x + x + 2 Decrease x term by 3 5x4 + 2x2 - x + (-5x4-x2-3x+2) = x2 - 4x + 2 Add 2 ___?__ = -5x4-x2-3x+2

11-2 Classifying Polynomials 11.2 11-2 Classifying Polynomials Objective: Classify polynomials by their degree or number of terms. Add and subtract polynomials. Big Idea: Polynomials are classified by their number of terms and by their degree. Goal: Understand the basic terminology of polynomials and the classification of polynomials by the number of terms or by their degree.

5x³y Term Review Coefficients of 1 are implied: x³ = 1∙x³ exponent variable coefficient Coefficients of 1 are implied: x³ = 1∙x³ Exponents of 1 are implied. An expression is considered to be simpler when it is written as x rather than 1x or . If there is no variable, the term is called a constant. It’s degree is zero. 5 = To find the total degree: add the exponents of all the variables

What is a polynomial? An expression that has no operations other than addition, subtraction, and multiplication by or of the variables. Every exponent must be a non-negative integer in a polynomial. Not polynomials: Fractional Exponents ex. Square Roots Absolute Values Terms divided by a Variable Terms with Negative Exponents (these are actually terms divided by a variable)

Polynomials vs. Not Polynomials 4x 5 3x²-5x³+2x-4 4y-3 3z³+6 |10-2y|

A. Classifying Polynomials by # of Terms Vocabulary: A. Classifying Polynomials by # of Terms 1) Monomial - an expression that can be written as a real number, a variable, or a product of a real number and one or more variables with non-negative exponents. ex. 6, x, 6xyz 2) Polynomial - an algebraic expression that is either a monomial or a sum of monomials 3) Binomial - a sum of two monomials ex. 5x + 3 4) Trinomial - a sum of three monomials ex. 5xy + 3x + 5 11.2

B. Classifying Polynomials by Degree: 5) Degree of a Monomial - the sum of the exponents of the variables in the monomial Ex. 6xyz  degree 3; -5x³  degree 3 6) Degree of a Polynomial - the highest degree of any of its terms after the polynomial has been simplified. 7) Linear - a polynomial of degree 1. ex. 3x + 1 8) Quadratic - a polynomial of degree 2. ex. 4x² 9) Cubic – a polynomial of degree 3 ex. 3x³ - 5 11.2

Monomial/Binomials/ Trinomials Name Number of Terms Example Monomial 1 -5x³ Binomial 2 4x²-5x Trinomial 3 4x²+2x-3 #-degree 4+

Degree Summary The degree of a polynomial with one variable is the exponent of the highest power of that variable. Degree Name Example First Linear 4x+5 Second Quadratic 5x²-4x+6 Third Cubic x³+6x Fourth Quartic Fifth Quintic 6+ No special name #-degree

Classifying Polynomials Can be done by both degree and number of terms Standard: State the degree first and then the number of terms

Classification of Polynomial Examples: State the degree and the type of polynomial. 5x³ 4x-3 3. 9-4x+2x³ 4. 5. 4x

6. 3x²-5x³+2x-4 7. 4y-3 8. 3z³+6 9. 25x²-100 10. 3x²-5x³+2x-4 11. 5

Write an example of each type of polynomial: 1. Linear binomial 2. Cubic trinomial 3. A 4th degree monomial 4. A quadratic trinomial

Descending Order Terms in a polynomial are listed in descending order of the exponents on the variable. As you go from left to right, the exponents go down in value. This is considered STANDARD FORM. NOTE: The value of the coefficient is not considered only the exponent value! Ex A. Write in Standard Form: 1 + 3y - 4y2 - 5y3 B. Write in Descending Form: -3 - 5x³ + y -10y² -5y3 - 4y2 + 3y + 1 – 5x3 – 10y2 + y - 3

Ascending Order Terms in a polynomial are listed in ascending order of the exponents on the variable. As you go from left to right, the exponents go up in value. Ex. Arrange in this polynomial in ascending order: 9y - 5y² - 4y³+ 1 1+ 9y - 5y2 - 4y3 + 1

Arranging terms in ascending/ descending order Simplify the polynomial first Commute the terms so that the exponents either increase or decrease from term to term. Always remember that the negative sign goes along with the term it proceeds. Remember if there is no exponent for a variable the exponent is a 1.

Arrange this polynomial in both ascending and descending order:

Standard form for more than one variable: There is no standard form, however, sometimes one variable is picked and the polynomial is written in decreasing powers of that variable. Example: a. Write in Standard Form as a polynomial in p. - q + 3p - 4p2 – p3q2 + p2q3 b. Write in Standard Form as a polynomial in q. p - q + 3p - 4p2 – p3q2 + p2q3 – p3q2 + p2q3 - 4p2 + 3p - q p2q3 - p3q2 - 4p2 + 3p - q

Adding and Subtracting Monomials Review Only add/subtract like terms. Remind me, what are like terms? Add coefficients keep exponent. Ex: 5x²-3x+4x²

Multiplying Monomials Review Multiply like bases. Add exponents, keep bases. Multiply coefficients Ex: (4xy)(3xy³)(-2x²)

Simplify. 3x2 + 5x2 4x3 - 2x2 2x6 - (4x6+7x6-9x6) (11x)(-3x) ab + ba

More Practice Examples: 1) Tell if the expression is a monomial. If so, identify its degree. If not tell why. a. 15x2 b. 156 c. d. ¾ x3 e. x3y4 f. 2x + 15x2 11.2 Yes – deg: 2 Yes – deg: 0 No – neg exponent Yes – deg: 3 Yes: deg: 7 No - binomial

2) Give the degree of each polynomial a. 1 + 3y + 4y2 + 5y3 b. p - q + 3p - 4p2 - p2q2 c. 4x4y-1 * + 7xy 11.2 Polynomial degree = 3 Polynomial degree = 4 = -6x3y2 + 7xy Polynomial degree = 5

List the following terms in ascending order.

Write a monomial with one variable whose degree is 5. 5) Write a monomial with one variable whose degree is 5. Write a monomial with two variables whose degree is 5. Write a trinomial with degree 5. 11.2 Sample answer : ½x5 Sample answer : -4x3y2 Sample answer : -4x3y2 +xy + 7

11-3 Multiplying a Polynomial by a Monomial 11.3 Objective: Multiply a polynomial by a monomial. Represent areas of figures with polynomials. Big Idea: To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and add the products. Goal: Apply the distributive property to multiply a polynomial by a monomial using area models to picture the porducts.

11.3 1) Give two equivalent expressions for the area pictured below. One is simplified. X + X + X = 3X X + 1+ 1+ 1+ 1 = 2X + 4 (3x) (2x + 4) = distribute X2 + X2 + X2 + x + x + x + x + x + x + x + x + x + x + x + x = (3x)(2x+4) = 6x2 + 12x (3x)(2x) + (3x)(4) =

11.3 2) Multiply k4(k2 - 16km) = k6 - 16k5m

11.3 3) Multiply -5y(y3 - 6y2 + 2y + 6) (-5y)(y3) – (-5y)(6y2) + (-5y)(2y) + (-5y)(6) Distribute -5y4 + 30y3 - 10y2 - 30y Multiply

4) Simplify 3x(x2 – 5) – (3x2 + 4x – 5) + 2(-2x2 – 4x) 11.3 3x3 - 15x - 3x2 - 4x + 5 - 4x2 - 8x Distribute 3x3 - 3x2 - 4x2 - 15x - 4x - 8x + 5 Commutative prop 3x3 - 7x2 - 27x + 5 Simplify

5) Draw boxes that would represent: (2x)(3x+2) = 6x2 + 4x 11.3 x + x + x + 1 + 1 x +

11-4 Common Monomial Factoring 11.4 11-4 Common Monomial Factoring Understand factoring as the reverse process of multiplication, concentrating on common monomial factors and their applications to the division of a polynomial by a monomial.

Trivial Factors – 1 and itself. Definitions: Factoring – the expression of a single monomial as the product of two or more factors. Trivial Factors – 1 and itself. Greatest Common Factor – the product of the gcf of the coefficients and the gcf of the variables. Factorization – the result of factoring a polynomial. Prime polynomials – monomials and polynomials that cannot be factored into polynomials of a lower degree. Complete factorization – When there are no common numerical factors in the terms of any of the prime polynomials. Ex 6x+12 factored completely is 6(x+2) 11.4

Unique Factorization Theorem for Polynomials 11.4 Unique Factorization Theorem for Polynomials Every polynomial can be represented as a product of prime polynomials in exactly one way, disregarding order and integer multiples. Factoring is the process of expressing a given number or expression as a product. The factored expression is always equivalent to the original polynomial.

NOTE: No terms should be lost in this process! 11.4 Steps for Factoring Find GCF for all terms. Divide all terms by GCF. Write the answer as a product of the GCF and the quantity of the remaining factors. NOTE: No terms should be lost in this process! The number of terms in the parenthesis should be the same as the original polynomial!

1) What are the factors of 8x3? 11.4 Factor the coefficient 8 1, 8 2, 4 Factor the variable(s) x3 x x2 Use all the individual factors and then combine them. 1, 2, 4, 8, x, x2, x3, 8x, 8x2, 2x, 2x2, 2x3, 4x, 4x2, 4x3 2) What is the greatest common factor between 8x3 and 12xy2 4x

= 4x (x4 + 3x2 + 2) Simplify all terms inside the parenthesis. 11.4 3) Factor 4x5 + 12x3 + 8x GCF of all 3 terms is 4x Factor/Divide the GCF from all terms. = 4x (x4 + 3x2 + 2) Simplify all terms inside the parenthesis.

4) Simplify 11.4 Find GCF of ALL terms top & bottom And factor it out…. Simplify fraction : Cancel Final answer

11.4 5) Illustrate the factorization of 4x2 + 12x by drawing a rectangle whose sides are the factors. = 4x (x + 3) x + 1+1+1 x +

11-5 Multiplying Polynomials 11.5 11-5 Multiplying Polynomials Objectives: Multiply polynomials having two or more terms. Represent areas and volumes of figures with polynomials.

11.5 Property: The Extended Distributive Property - To multiply two sums, multiply each term in the first sum by each term in the second sum.

Property: Multiplying two binomials The FOIL algorithm: First Outside Inside Last (a + b)(c + d) = 11.5 ac + bd + ad + bc

1) Multiply (3n + 10)(7n + 2) = (3n)(7n) + (3n)(2) + (10)(7n) 11.5 1) Multiply (3n + 10)(7n + 2) = (3n)(7n) + (3n)(2) + (10)(7n) + (10)(2) = 21n2 + 6n + 70n + 20 multiply = 21n2 + 76n + 20 simplify

2) Multiply a binomial by a trinomial – remember: The Extended Distributive Property – To multiply two sums, multiply each term in the first sum by each term in the second sum. 11.5 (n - 5)(2n2 - 3n + 7) = (n)(2n2) - (n)(3n) + (n)(7) - (5)(2n2) + (5)(3n) - (5)(7) = 2n3 - 3n2 + 7n - 10n2 + 15n - 35 multiply = 2n3 - 13n2 + 22n - 35 simplify

3) Multiply a trinomial by a trinomial 11.5 3) Multiply a trinomial by a trinomial (w2 + 4w + 6)(w2 + w + 1) = w2(w2) + w2(w) + w2(1) + 4w(w2) + 4w(w) + 4w(1) + 6(w2) + 6(w) + 6(1) = w4 + w3 + w2 + 4w3 + 4w2 + 4w + 6w2 + 6w + 6 multiply = w4 + 5w3 + 11w2 + 10w + 6 simplify Remember standard form

11.5 4) (2a + 10b + 1)(4a + b+ 1) = 2a(4a) + 2a(b) + 2a + 10b(4a) + 10b(b) + 10b + 4a + b + 1 = 8a2 + 2ab + 2a + 40ab + 10b2 + 10b + 4a + b + 1 = 8a2 + 6a + 42ab + 10b2 + 11b + 1 simplify

5) a. Express the area as the sum of the tiles. 11.5 5) a. Express the area as the sum of the tiles. b. Express the area as the length * width c. What equality is shown? = 8x2 + 14x + 3 = (2x + 3)(4x + 1) = 8x2 + 14x + 3 = (2x+3)(4x+1) x + x 1+1+1 x + 1 x2 x2 x x x x x 1 1 1

11-6 Special Binomial Products 11.6 11-6 Special Binomial Products Objectives: Apply two patterns of binomial multiplication, the square of a binomial and the difference of squares, to do arithmetic multiplication mentally and to illustrate how a knowledge of algebra can contribute to increased arithmetic proficiency. Multiply two binomials Expand squares of binomials Represent the square of a binomial as an area

1) Perfect Square Patterns: For all numbers a and b, Properties: 1) Perfect Square Patterns: For all numbers a and b, (a+b)2 = (a+b)(a+b) = a2 + 2ab + b2 (a-b)2 = (a-b)(a-b) = a2 - 2ab + b2 2) Difference of Two Squares Pattern: For all numbers a and b, (a+b)(a-b) = a2 - b2 11.6

1) The length of the side of a square is y + 7. a. Write the area of the square in expanded form. b. Draw the square and show how the expanded form relates to the figure. 11.6 Remember: (a+b)2 = a2 + 2ab + b2 (y+7)2 = y2 + 2(y)(7) + 72 (y+7)2 = y2 + 14y + 49 y +1+1+1+1+1+1+1

2) Expand (2n-5)2 3) Multiply (10n - 7)(10n + 7) 11.6 2) Expand (2n-5)2 Remember: (a-b)2 = a2 – 2ab + b2 (2n-5)2 = (2n)2 – 2(2n)(5) + (5)2 (2n-5)2 = 4n2 – 20n + 25 3) Multiply (10n - 7)(10n + 7) Remember: (a+b)(a-b) = a2 - b2 (10n+7)(10n-7) = (10n)2 - 72 (10n+7)(10n-7) = 100n2 - 49

Let w = mat width (and length) 4) An 8" by 8" square photograph is to be surrounded by a square mat with width w. Sketch the photo and mat. Express the area of the mat that shows as a product of 2 binomials. 11.6 Let w = mat width (and length) Mat Area = w2 - 82 Mat Area = (w+8)(w-8)

5) Compute 512 in your head. 512 = (50 + 1)2 11.6 512 = (50 + 1)2 Remember: (a+b)2 = a2 + 2ab + b2 512 = (50+1)2 = 502 + 2(50)(1) + 12 = 2500 + 100 + 1 = 2601

Review for Final Solve for x 3x + 4y – z = 4x + 7y Solve for y

11-7 Permutations Objectives : 11.7 11-7 Permutations Objectives : Find the number of permutations of objects without replacement. Understand factorial notation.

11.7 Permutation: An arrangement where order is important. Example P(14,4) = 14 • 13 • 12 • 11 Factorial: n! means the product of all counting numbers from n down to 1. Example 6! = 6 • 5 • 4 • 3 • 2 • 1 = 720 P(6,6) = 6!

11.7 Ex 1) There are 10 players on a softball team. In how many ways can the manager choose three players for first, second, and third base? Use the Fundamental Counting Principle (8-1) number of possible players for first base number of possible players for second base number of possible players for third base total number of possible ways x = 10 9 8 720 x = = Permutations of 10 players chosen 3 at a time = P(10,3) Book language: permutations chosen from 10 of length 3 Answer: There are 720 different ways the manager can pick players for first, second, and third base.

11.7 (Your turn) Ex 2) There are 15 students on student council. In how many ways can Mrs. Sommers choose three students for president, vice president, and secretary? Answer: P(15,3) = 15 · 14 · 13 = 2,730 Ex 3) Find the value of P(7,5) 7 · 6 · 5 · 4 · 3 = 2520 7 things of length 5.

Ex 4) Evaluate 5! Read “5 factorial” 11.7 Ex 4) Evaluate 5! Read “5 factorial” 5! = 5 · 4 · 3 · 2 · 1 = P(5,5) = 120 Ex 5) Evaluate = = 90·89·88 = 704880 Ex 6) Evaluate = =

1’s column digit: 2,4,6,8, 4 available digits 11.7 Ex 7) How many ways can you make an 7 digit number if you only use the digits 1-9 and you must have an even number, and no number can be used twice? Digits to be used: 1’s column digit: 2,4,6,8, 4 available digits 1st digit: 1-9 less 1’s col digit 8 available digits 2nd digit: 1-9 less 2 digits 7 available digits … This is a permutation for the 1st 6 digits P(8,6) 7th digit must be even: there are 4 digits that would result in an even number: 2, 4, 6, 8 P(8,6) · 4 = 80,640

Chapter 11 Review 1) Write a polynomial in standard form. 2) Write a three variable polynomial with 5 terms and a degree of 5. Ex: 4x2 + 3x + 6 Ex: 4x2y2z + xz + xy + 3x + 6

3) Simplify. a. -3x(4x2 + 2x -5) b. (y - 4)(y + 2) (3r + 1)(3r - 1) d. (4x2 - 3x + 2) + (2x2 - 2) Rev 11 -12x3 - 6x2 + 15x y2 - 2y - 8 9r2 - 1 difference of 2 squares 6x2 – 3x simplify

e. (5k - 2j) - (2k - 3j + 5) (4t - 2)(t4 + 3t2 + 4) 4) Factor Rev 11 e. (5k - 2j) - (2k - 3j + 5) (4t - 2)(t4 + 3t2 + 4) 4) Factor 36x2 + 12x + 6 ½ x2y + 4xy2 + xy 3k + j – 5 simplify 4t5 - 2t4 + 12t3 - 6t2 + 16t - 8 6(x2 + 2x + 1) xy(½x + 4y + 1)

5) Know the Perfect Square Patterns and Difference of Two Squares Pattern. Rev 11 1) Perfect Square Patterns: (a+b)2 = (a+b)(a+b) = a2 + 2ab + b2 (a-b)2 = (a-b)(a-b) = a2 - 2ab + b2 2) Difference of Two Squares Pattern: (a+b)(a-b) = a2 - b2 Expand: (7 + x)2 b) (2n – 6)2 c) (4z + 3a)(4z – 3a) x2 + 14x + 49 4n2 – 24n + 36 16z2 – 9a2

6) State the multiplication shown in a picture. a. Rev 11 6) State the multiplication shown in a picture. a. b. (x+2) (x+1) = x2 + 3x + 2 x · x (x+y) = x3 + x2y

7) Investment situation. Each birthday from age 15 on, Joan has received $75 from her grandfather. She puts the money into a savings account with a yearly scale factor of p and does not make any withdrawals or additional deposits. a. Write an expression for the amount Joan will have in the account on her 18th birthday. b. If the bank pays 6% interest per year, how much will Joan have on her 18th birthday. Rev 11 75p3 + 75p2 + 75p + 75 75(1.06)3 + 75(1.06)2 + 75(1.06) + 75 = $328.10