Precalculus Chapter P.1 Part of 3 Mr. Chapman Manchester High School
Algebraic Expressions Evaluating Algebraic Expressions Using the Basic Rules and Properties of Algebra
Definition of an Algebraic Expression: A collection of letters (variables) and real numbers (constants) combined using: Addition Subtraction Multiplication Division Exponentiation
Examples: 5x x 3 1 term terms 7x y terms 1 term 4 x Terms of an Algebraic Expression: Individual terms are separated by addition or subtraction signs. How many terms in each of the algebraic expressions above?
Consider the polynomial: x 5x 8 How many terms? 3 variable terms: constant terms: 1 Coefficients are the numerical factor (multiplier) in a variable term.? 1 x ( 5) x 8
Challenge Question! 5x 8 Given the polynomial What is the number 8? A: a coefficient C: both A & B B: a constant term D: not A or B x Does it multiply a variable? Is it a term by itself? YES!! No
Algebraic Expressions Evaluating Algebraic Expressions Using the Basic Rules and Properties of Algebra
To evaluate Algebraic Expressions: Replace the variables with their assumed or desired numerical value. This is called the Substitution Principle.
Examples: Value of Expr. Expression Variable Substitution Value 3x 5 x 3 3(3) 5 95 4 3 x x 1 x 1 3( 1) ( 1) 1 3 1 0
Algebraic Expressions Evaluating Algebraic Expressions Using the Basic Rules and Properties of Algebra
Four Arithmetic Operations: Addition (+) Subtraction (-) Multiplication ( or ) Division ( or ) a b a ( b) 53 5 ( 3) Additive Inverse Primary Operations Addition Multiplication a a 1 b b Multiplicative Inverse 1 1 4 Secondary (Inverse) Operations Subtraction Division 1 4
Let a, b, and c be real numbers, variables, or expressions. 5, x, 3x Rules and Properties of Algebra are only given for the Primary Operations. They are considered to also apply to secondary (inverse) operations by extension.
Commutative Property of Addition a b b a 4x x x 4x Memory Tip: Commute means travel (as in commute to work ). Note that the terms in this property exchange places or travel.
Commutative Property of Multiplication ab ba (4 x) x x (4 x) Memory Tip: Commute means travel (as in commute to work ). Note that the terms in this property exchange places or travel.
Associative Property of Addition ( a b) c a ( b c) ( g b) ( x( y 5) w ) x g x ( b (5 y x w) ) Memory Tip: Terms do not commute or travel. (That is, their order stays the same!) Terms change who they associate with (are grouped with) by using parentheses.
Associative Property of Multiplication ( ab) c a( bc) ( x3y)8 x(3y 8) Memory Tip: Terms do not commute or travel. (That is, their order stays the same!) Terms change who they associate with (are grouped with) by using parentheses.
Distributive Property a( b c) ab ac 3x(5 x) 3x53xx Memory Tip: Both addition and multiplication involved. The term outside the parentheses gets distributed to (or shared with) every term inside the parentheses. (Like treats!)
Distributive Property ( a b) c ac bc ( y 8) y y y 8 y Memory Tip: Both addition and multiplication involved. The term outside the parentheses gets distributed to (or shared with) every term inside the parentheses. (Like treats!)
Distributive Property ( a b) c ac bc ( y 8) y y y 8 y ( y 8) y y y 8 y Memory Tip: Both addition and multiplication involved. The term outside the parentheses gets distributed to (or shared with) every term inside the parentheses. (Like treats!)
Additive Identity Property a 0 a 5y 0 5y Memory Tip: In math, an identity is true for all values. Identity also means the state of being the same. In this case the value of a is unchanged.
Multiplicative Identity Property a1 a 4x 1 4x Memory Tip: In math, an identity is true for all values. Identity also means the state of being the same. In this case the value of a is unchanged.
Additive Inverse Property a ( a) 0 5x 3 ( 5x 3 ) 0 Memory Tip: The inverse of addition is subtraction. Subtraction is adding the same value but of the opposite sign.
Multiplicative Inverse Property 1 a 1, a 0 a 1 ( x 4) 1 x 4 Memory Tip: The inverse of multiplication is division. Division is multiplying by one over something.
Homework: Pg. 10: 67-111 Odd