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Using Algebra Tiles to Represent Expressions Algebra tiles are shaped tiles that can be used to represent expressions and to help solve equations (equations have an equal sign, expressions don’t). There are 3 different tiles that are used: Unit tile – used to represent the value ‘1’ X tile – used to represent the variable ‘X’ X2 tile – used to represent the variable ‘X2’ These tiles either come in two different colours, or have different colours on each side. One colour represents positive values, and the other represents negative values. Example: Show x2 + 4x – 5 using algebra tiles. Solution: Working with Exponents ‘Powers’ are a shorter way to write repeated multiplication. They are written with a base and an exponent. The base tells us the factor to multiply by. The exponent tells us how many times to complete the multiplication. For example: Exponent 45 = 4 x 4 x 4 x 4 x 4 Base Tips: When solving equations, remember to follow BEDMAS rules. Some calculators have x2 buttons Some calculators also have yx buttons for other exponents. Exponent Rules To help us solve complicated equations with exponents, there are a set of rules we can follow: Product Rule: When multiplying powers with the same base, add the exponents and write the product as a single power. Xa x Xb = Xa+b Example: 22 x 25 = 22+5 = 27 But how do we know it works? Try it the long way: 22 x 25 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 27 Quotient Rule: When dividing powers with the same base, subtract the exponents and write the quotient as a single power. Xa Xb = Xa-b Example: 46 44 = 46-4 = 42 Try it the long way: (cancel out the pairs) 46 44 = 4x4x4x4x4x4 4x4x4x4 = 42 Power of a Power Rule: A power of a power can be written as a single power by multiplying the exponents. (Xa)b = Xa x b Example: (32)4 = 32x4 = 38 Try it the long way: (32)4 = (3 x 3) x (3 x 3) x (3 x 3) x (3 x 3) = 38 Communicating With Algebra – Polynomials A polynomial is an expression made up of one or more terms, connected by addition or subtraction. They are classified based on the number of terms they have: One term Two terms Three terms monomial binomial trinomial x, 3y, 4a, etc. 2x – 3, 5b + 2, etc. 4a + 2b – 7, 2x – 3y + 4, etc. The terms in a polynomial are made up of 2 different parts: the variable and the coefficient. The variable is represented with a letter, and includes any exponents that are attached to it. The coefficient is a number that is part of a term. When there is a coefficient without a variable, it is called a constant. Example: 2b2 the variable is b2 and the coefficient is 2 3x the variable is x and the coefficient is 3 6 the constant is 6 When we are describing polynomials we can also describe them based on the degree of a specific term, or of the entire polynomial. Degree of a term: the sum of the exponents in a single term. Example: 2b2c3 2+3=5 The degree of the term is 5. Degree of a polynomial: the degree of the term with the highest degree within the polynomial Example: 3x2 + 6x – 2 3x2 is the term with the highest degree. The degree of the term is 2 The polynomial is a second degree polynomial. Polynomials – Collecting Like Terms When we work with complicated polynomials it is good to simplify the expression to make it easier to use. When we simplify an expression we gather the like terms together. We decide if terms are alike or not based on the variable. For example: 6x, x, 7x, -2x, and –3x are all like terms. -x2, 4x2, and 6x2 are all like terms 4, 5, 8, and 2 are all like terms 6x and –4x2 are not like terms, because the variables are different. Once the like terms are grouped together you can add or subtract them to make simpler expression. Example: 300 + 4x + 2 x + 150 = = 300 + 150 + 4x + 2x 450 + 6x Add and Subtract Polynomials To add two or more polynomials together, we remove the brackets, and then collect like terms. Example: (4x + 3) + (7x + 2) = 4x + 3 + 7x + 2 = 4x + 7x + 3 + 2 = 11x + 5 Remember: the sign (+ or -) must go with the term Example: (0.5v2 + 2v) + (-2.4v2 – 3v) = 0.5v2 + 2v - 2.4v2 – 3v = 0.5v2 - 2.4v2 + 2v – 3v = - 1.9 v2 – v the “-“ stays with the term if the coefficient is 1, we leave it off To subtract two polynomials, we add its opposite. Example: (3y + 5) – (7y – 4) = (3y + 5) + (-7y + 4) = 3y + 5 – 7y + 4 = 3y – 7y + 5 + 4 = - 4y + 9 (-7y + 4) is the opposite of (7y – 4) Polynomials – The Distributive Property When we multiply a polynomial by a number (like ‘5’) or a variable (like ‘x’), we use a rule called the Distributive Property. a(x + y) = ax + ay Examples: 3(x + 2) = 3x + 6 3 x x = 3x, 3 x 2 = 6 -6(3b – 3) = -18b + 18 -6 x 3b = -18b, -6 x –3 = + 18 x(x – 3) = x2 – 3x X x X = X2, X x 3 = 3X Notes: Sometimes a polynomial will include multiplication. We must do the multiplication before simplifying. If we have brackets within brackets, solve the inner brackets first, including any multiplication. Name: ________________________________ Date: _____________________ Class: _____ MPM 1D Chapter 3 Test – Polynomials Knowledge & Understanding: _____ / 24 Application: _____ / 9 Communication: _____ / 7 Thinking: _____ / 6 Multiple Choice (10 Marks K) Write the letter of the correct answer on the line beside the question. _____ 1. Which algebraic statement best represents this model? a) 3x + 3 b) 3 c) 3x2 d) 3x _____ 2. What is the simplified form of the expression w2 x w4 x w? a) w5 b) w7 c) w8 d) w9 _____ 3. Which of the following includes three like terms? a) 3x 2x -x2 b) 4a 4b 4c c) 3y -y 8y 2 d) 3m n 2mn 4mn2 _____ 4. 4x2 + 3x – 14 is an example of a… a) term b) monomial c) binomial d) trinomial _____ 5. What is the degree of the term 3v3m? a) 1 b) 3 c) 6 d) 9 _____ 6. What is the degree of the polynomial x2 + 6x – 2? a) 1 b) 2 c) 3 d) 4 _____ 7. 3x – 5 + 2x + 3 can be simplified to… a) 3x b) 5x – 2 c) x – 2 d) 5x + 2 _____ 8. -2m(3m – 1) simplifies to… a) -6m2 + 2m b) -6m2 – 1 c) 6m2 – 2m d) -6m2 – 2m _____ 9. In the expression 6x + 4, the 4 is referred to as the… a) term b) constant c) coefficient d) variable ____ 10. In the term 45, the 5 is referred to as the… a) exponent b) power c) base d) raised Short Answer (14 Marks K) Complete all work in the space provided. Show your steps. 11. Evaluate/Simplify as appropriate. (6 Marks K) a) (-3)4 b) 23 x 22 x 24 c) 2k + 5m – k – 6m d) 3(y – 7) e) (2/3)4 f) -4k(2k + 6) 12. Simplify. (8 Marks K) a) (5x – 3) + (2x + 7) b) (y2 – 3y) – (2y2 – 5y) c) 2(q – 5) + 4(3q + 2) d) -3(2m – 6) – (8 – 6m) Full Response/Problems (9 Marks A, 7 Marks C, 6 Marks T) Complete all questions in the space provided. Show all work, and provide explanations. Use the back of the page if you need more space. 13. $50 is put into a bank account that pays interest so that the amount in the account will grow according to the expression 50(1.03)n, where n is the number of years the money is left in the account. How much money will be in the account in… a) 10 years (4 Marks A, 2 Marks C) b) 20 years 14. Sarah and Martin are planning to paint the lines on the school soccer field. Sarah says, “The formula for perimeter is P = 2l + 2w.” Martin replies, “That’s not right, the formula for perimeter is P = 2(l + w).” Who is right? Is it possible that they are both right? How do you know? (2 Marks A, 1 Mark C) Find at least one other way to write a valid formula for the perimeter of a rectangle, and explain why it works. (1 Mark T, 1 Mark C) 15. The sum of the perimeter of two shapes is represented by 13y + 4x. The perimeter of one shape is represented by 4y – 2x. What is the perimeter of the second shape? (3 Marks A, 1 Mark C) 16. Three teachers team up to edit a Math textbook. The publisher pays each according to their own contract: Teacher Mr. Law Ms. Vorobej Mr. Scharf Flat Fee ($) 4000 2000 1000 Royalty ($ for n books sold) none 4n 6n Create a simplified expression that represents the total the publisher must pay the teaching team. (4 Marks T, 1 Mark C) Determine the total payout if the textbook sells 5000 copies. (2 Marks T, 1 Mark C)