Math 017 Materials With Exercises

FactorsAlgebraic expressions that are multiplied are called factors.The expression 4 mn has factors 1, 4, m, n and any combination of those, like 4 m , 4 n , and of course,22224 mn . In ab , the expressions a and b are called factors but 1, b, b , ab, ab are also factors of thisexpression. During our study we will be talking about explicit factors. Explicit factors of 4 mn are 4,m and n, i.e. the factors that are separated by the multiplication sign (displayed or not displayed) of an22expression. Explicit factors of ab are a and b . The expressions 2x 3 and x 4 are explicit factorsin ( 2x 3)( x 4).Example 3.1 List all terms of the following expressions.a) 3 y zb)x 3x2 4( z 1)2Solution:a) The terms are 3, y, z . Remember that signs are always a part of terms.We list y as a term (y is preceded by the minus sign).xb) The terms are 3 2 2 x x ,4( z 1), . Notice the minus sign in 3x and ,22and that the expressions 4(x 1)should be viewed as one term.Example 3.2 List all explicit factors of the following multiplication.a) 5 abb) 3(a 1)xSolution:a) Since 5 ab 5 a b, the explicit factors are 5 , a, b.b) Since 3 ( a 1)x 3( a 1)b, the factors are; 3,( a 1)and x . Noticethat when listing factors, the expressions in parentheses are treated as one “unsplitable”expression.Algebra is an abstract generalization of arithmetic, where numbers are ‘replaced’ with variables. Thelaws that are true for numbers also hold for algebraic expressions (recall, algebraic expressions aremerely symbolic representations of numbers).We will discuss some of the laws in the context of equivalent expressions.Commutative property of addition: rearranging terms results in equivalent expressionsWe know that 3 5 and 5 3 are both equal to the same number, 8. It is because the result of additiondoes not depend on the order of numbers that are being added. This property is called the commutativeproperty of addition. Remember, subtraction does not have this property: 5 3 35 . But, if we viewsubtraction as the addition of the opposite number, we get 5 3 5 ( 3) 3 5 . With the use ofvariables, we can express the above ideas in a general form (without the use of specific numbers). Forany value of x and y,25