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Topic 5

Topic 5. Simplifying expressions and Properties of Operations. 7.5.1 algebraic expressions. To evaluate an algebraic expression you replace each variable with its numerical value, then use the order of operations to simplify.

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Topic 5

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  1. Topic 5 Simplifying expressions and Properties of Operations

  2. 7.5.1 algebraic expressions • To evaluate an algebraic expression you replace each variable with its numerical value, then use the order of operations to simplify. • PEMDAS: Parenthesis, exponents, multiplication/division, addition/subtraction

  3. Example 1: w=2 2w = 2*2 = 4 Example 3: n=7 (n+2)2 / 3 = (7+2)2/ 3 = (9)2/ 3 = 81 / 3 = 27 Example 2: p=9 3.3p + 2 = 3.3*9 + 2 = 29.7 + 2 = 31.7

  4. 7.5.4 the distributive property • To multiply a sum or difference by a number, multiply each term inside the parentheses by the number outside the parentheses. • a (b + c) = ab + ac • a (b - c) = ab - ac

  5. 7.5.5 Simplify Algebraic Expressions • When a plus or minus sign separates an algebraic expression into parts, each part is called a term. • The numerical factor of a term that contains a variable is called the coefficient of the variable. • A term without a variable is called a constant. • Like terms contain the same variables to the same powers, such as 3x2 and 2x2. • An algebraic expression is in simplest form if it has no like terms and no parentheses.

  6. Example 2 • (4x –y)9 = • [4x + (-y)]9 = • (4x)9 + (-y)9 = • 36x + (-9y) = • 36x –9y Example1 5 (x + 3) = 5 · x + 5 · 3 = 5x + 15

  7. 7.5.6 Add linear expressions Linear expression: an algebraic expression where the variable is raised to the first power (not square/cubed/etc) and the variables are not multiplied or divided. Examples of linear expressions: • 5x • 3x + 2 • x – 7 Expressions that do not follow these rules are called nonlinear expressions. Examples of nonlinear expressions: • 5mn • 3x3 + 2 • x4 – 7 Directions for completing problems: The plus (positive) sign in between the parentheses means that everything inside the following parenthesis keep the sign it has.

  8. Example1 (5x + 7) + (x + 2) = 5x + 7 + x + 2 = Because we are adding, rewrite without the parentheses 5x + x + 7 + 2 = Rearrange like terms together 6x + 9 Combine like terms Example 2 (– 6x + 3) + (x – 7) = -6x + 3 + x – 7 = Because we are adding, rewrite without the parentheses -6x + x + 3 – 7 = Rearrange like terms together -5x – 4 Combine like terms

  9. 7.5.7 Subtract linear expressions • The negative sign before a parenthesis applies to every term inside the following parenthesis. (You must switch the sign of each of the terms inside that next parenthesis.)

  10. Example1 (6x + 7) – (2x + 2) = 6x + 7 – 2x – 2 = Because we are subtracting, switch the sign of each term in the 2nd parenthesis 6x – 2x + 7 – 2 = Rearrange like terms together 4x + 5 Combine like terms Example 2 (2x – 3) – (x – 2) = 2x – 3 – x + 2 = Because we are subtracting, switch the sign of each term in the 2nd parenthesis 2x – x – 3 + 2 = Rearrange like terms together x – 1 Combine like terms

  11. 7.5.8 Factor Linear Expressions • A monomial is a number, a variable, or a product of a number and one or more variables. • Monomials: 25, x, 40x • Not monomials: x + 4, 40x + 120 • To factor a number means to write it as a product of its factors. A monomial can be factored using the same method you would use to factor a number. • The GCF (greatest common factor) of two monomials is the greatest monomial that is a factor of both. • Reminder: prime numbers are numbers that can be factored only by 1 and the number itself.

  12. Find the GCF of each pair of monomials Example1 4x, 12x 4x = 2 x 2 xxFind the prime factors of 4x 12x = 2 x 2 x 3 x xFind the prime factors of 12x Find the common factors 2 x 2 x x = 4x Combine the prime factors to find the GCF Example 2 18a, 20ab 18a = 2 x3 xaFind the prime factors of 18a 20ab = 2 x 2 x 5 x a x b Find the prime factors of 20ab Find the common factors 2 x a = 2a Combine the prime factors to find the GCF

  13. 7.5.8 Factor linear expressions (continued)

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