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MFM1PI: Foundations of Mathematics Unit 4: Working with Algebra Daily Notes Day 1 2 Date Topic Intro to Algebra & Like Terms 3 Multiplying and Dividing Monomials 4 Combining Operations 5 Distributive Property Part 1 6 Distributive Property Part 2 7 Algebra Review 8 Unit Test Day 1: Intro to Algebra Date: Algebraic expressions use letters and numbers: 2 4x – 3y + z + 5 Definitions: An algebraic expression like the one above is called a _______________. This polynomial has four parts called __________, separated by + or –. 4x2, 3y, and z are called ___________ terms. The “5” is also a term on its own, and is called a ____________ term. The letters in a polynomial are called ____________. The variables in this polynomial are ____ & ____ & ____. The numbers are called different things depending on what they do: The “5” is a ____________ term because it is a non-variable term. The “3” and “4” are _____________ because they multiply variables. The “2” is an _____________ affecting the variable ____. The coefficient of “z” is ____. The exponent of “z” is ____. Terms with identical _____________ (including __________) are called _________ terms. They can be combined by adding or subtracting. Polynomials can be named according to the number of terms they have: # of Terms 1 2 3 Name Example Day 1: Adding Polynomials Date: If you want to add polynomials together, you can only put like terms together. The steps are: Mark like terms using underlines, boxes, etc. Add the coefficients of the like terms, keep the variable. Examples: 4x + 3x + 9x 2x + y + 3x + 5y 3x2 + 7x – 4x2 + 9 More Examples: (4x2 - x + 5) + (9x2 - 2x – 6) (8x2 + 7x + w) + (w + 2x – 3x2) Day 2: Subtracting Polynomials Date: If you want to subtract polynomials, you can only put like terms together. The steps are: Distribute “–“ signs over any brackets. Mark like terms using underlines, boxes, etc. Add/subtract the coefficients of the like terms, keep the variable. Examples: (6x2 - 2x + 15) - (9x2 + 2x – 6 ) (2x2 + 8x + y) - (y + 3x – 3x2) Examples: (4x2 - x + 5) - (9x2 - 4x + 1) (8x2 + 7x + w) - (2w + 2x – 3x2) Day 3: Multiplying and Dividing Monomials When multiplying and dividing monomials, the exponents on the variables can change. (Whereas with adding and subtracting, the exponents never change). You will have to remember the first two power rules from the exponents unit. Steps for multiplying monomials Multiply the coefficients Add the exponents on the variable Date:__________________ When multiplying powers, Add the exponents (x5)(x3) = x8 When dividing powers, Subtract the exponents x5 ÷ x3 = x2 Steps for dividing monomials Divide the coefficients, or put the fraction in lowest terms Subtract the exponents on the variable Examples 3x7 × 4x3 = (-2x)(9x5) = -2y7 · 3y7 · (-4y7) = 4x10 ÷ 2x-2 = 30𝑥 10 = 12𝑥 7 −12𝑥 7 4𝑥 5 = Day 4: Combining Operations Date:__________________ Examples 5) 4𝑎𝑏7 2𝑎5 = 6) 2𝑘ℎ3 𝑗 4 4𝑘ℎ2 𝑗 = Day 5: Distributive Property Part 1 Date:__________________ Any polynomial can be multiplied by a monomial. Each term in the brackets will be multiplied by the monomial on the outside. This kind of multiplying is called distribution. How it works: BE[DM][AS] Method Distributive Method 10(5 – 3) 10(5 – 3) = 10(2) = 50 – 30 = 20 = 20 Examples Day 6: Distributive Property Part 2 Date:__________________ Instead of having just a number on the outside, we can also have a variable. You’ll have to remember to use your exponent rules for multiplying with a variable. Examples Expand Expand & Simplify Day 7: Algebra Review Date: Algebra Graphic Organizer Operation Steps Add Collect Like Terms Combine Coefficients Subtract Distribute “-“ with brackets Collect Like Terms Combine Coefficients Multiply Multiply Coefficients Multiply Variables o (Add Exponents) Divide Divide Coefficients Divide Variables o (Subtract Exponents) Distributive Property Multiply the “outside” term by each of the “inside” terms Simplifying Expressions Do Distributive Property First Collect Like Terms Combine Coefficients Example