Page 1 of 6 ALGEBRAIC EXPRESSION A cobination of ordinary nubers, letter sybols, variables, grouping sybols and operation sybols. Nubers reain fixed in value and are referred to as constants. Letter sybols represent nubers that are fixed in value, but are unspecified. Variables are letter sybols representing single nubers or sets of nubers. 5 5xy z Exaples: 5, x 6, x 5 6, x 5xy y, a b, a c TERM Consists of products and quotients of ordinary nubers and letters that represent nubers. Grouped sybols are considered as a single nuber. 7 Exaples: 6x y, 5x / y, x x y y 1/ a x,, MONOMIAL An algebraic expression consisting of only one ter. Monoials are soeties siply called ters. Exaples: 7x y, xyz, x / y BINOMIAL An algebraic expression consisting of two ters. Exaples: x y, x xyz TRINOMIAL An algebraic expression consisting of three ters. Exaples: x 5x, x 6y z, x xy / z x z MULTINOMIAL An algebraic expression consisting of ore than one ter. Exaples: 7x 6y, x 6x y 7xy 6, 7x 5x / y x / 16 COEFFICIENT One factor of a ter is said to be the coefficient of the rest of the ter. Exaples: in the ter 5x y, 5x is the coefficient of y, 5y is the coefficient of and 5 is the coefficient of x y The nueric factor of a ter is the nueric coefficient and is often referred to siply as the coefficient. Exaple: in the ter 5x y, 5 is the nueric coefficient. 7 x, Copyright 001
Page of 6 LIKE TERMS or SIMILAR TERMS Ters which differ only in nueric coefficients. Exaples: 7 xy and xy are like ters, and 1 x y and x y are like ters. Note: 7 a b and a b are unlike ters. Two or ore like ters in an algebraic expression ay be cobined into one ter. Exaple: 7x y x y x y ay be cobined as 5 x y INTEGRAL and RATIONAL TERMS Ters consisting of a) positive integral powers of literal nubers ultiplied by a factor not containing the letters -orb) no literal nubers at all are integral and rational. 6 Exaples: 6x y, 5y, 7, x, x y are integral and rational in the letters present. Note: x is not rational in x and / x is not integral in x POLYNOMIAL A onoial or ultinoial in which every ter is integral and rational in the literals. Exaples: x y 5x y, x 7x x 5x, xy z, x Note: x / x and y are not polynoials. DEGREE of a MONOMIAL The su of all the exponents in the literal part of the ter. Exaple: the degree of x y z is 1 or 6 Note: the degree of a constant such as 6, 0, and is zero. DEGREE of a POLYNOMIAL The degree of the ter having the highest degree and a non-zero coefficient. 5 Exaple: 7x y xz x y has ters of degree 5, 6, respectively hence, the degree of the polynoial is 6 Copyright 001
Page of 6 SYMBOLS of GROUPING Parentheses ( ), brackets [ ], and braces { } are often used to show that the ters contained in the are considered as a single quantity. Exaple: two algebraic expressions 5x x y and x y ay be cobined as a su: 5 x x y x y or a difference: 5 x x y x y or a product: 5 x x y x y REMOVAL of SYMBOLS of GROUPING If a + sign precedes a sybol of grouping, this sybol ay be reoved without affecting the ters contained. Exaple: x 7 y xy x x 7y xy x If a sign precedes a sybol of grouping, this sybol ay be reoved only if the sign of each ter contained is changed. Exaple: x 7 y xy x x 7 y xy x If ore than one sybol of grouping is present, the inner ones are to be reoved first. x x x 5y x x x 5y x x x 5 Exaple: y ADDITION of ALGEBRAIC EXPRESSIONS Achieved by cobining like ters. In order to accoplish this addition, the expressions ay be arranged in rows with like ters in the sae colun; these coluns are then added. Exaple: to add 7x y xy, x y 7xy and xy 5x 6y write: and add 7x y xy x y 7xy 5x 6y xy 5x 5y 5xy SUBTRACTION of ALGEBRAIC EXPRESSIONS Achieved by changing the sign of every ter in the expression that is being subtracted and adding this result to the other expression. Exaple: to subtract x 5xy 5y fro 9x xy y write: and add ( 9x xy y = x 5xy 5y ) = 9x xy y x 5xy 5y 7x xy 8y Copyright 001
Page of 6 MULTIPLICATION of ALGEBRAIC EXPRESSIONS To ultiply two or ore onoials, use the laws of exponents, the rules of signs, and the coutative and associative laws of ultiplication. Exaple: to ultiply x y z, x y and xy z write: x y z x y xy z rearrange: x x x y yy z z 7 8 and ultiply x y z To ultiply a polynoial by a onoial, ultiply each ter of the polynoial by the onoial and cobine results. Exaple: to ultiply xy x xy and 5x y write: 5 x y xy x xy distribute: 5 x y xy + 5 x y x + 5 x y xy 5 5 6 and ultiply 15x y 0x y 10x y To ultiply a polynoial by a polynoial, ultiply each of the ters of one polynoial by each of the ters of the other polynoial and cobine results. Exaple: to ultiply x 9 x and x x x 9 x write: distribute: x 9 x + x x 9 x ultiply: 9x 7 x + x 9x x and add x 6x 18x 7 DIVISION of ALGEBRAIC EXPRESSIONS To divide a onoial by a onoial, find the quotient of the nueric coefficients, find the quotients of the literal factors, and ultiply these quotients. To divide a polynoial by a polynoial a) Arrange the ters of both polynoials in descending (or ascending) powers of one of the letters coon to both polynoials. b) Divide the first ter in the dividend by the first ter in the divisor. This gives the first ter in the quotient. c) Multiply the first ter of the quotient by the divisor and subtract fro the dividend, thus obtaining a new dividend. d) Use the dividend obtained in c) to repeat steps b) and c) until a reainder is obtained which is either of degree lower than the degree of the divisor or zero. dividend reainder e) The result is written quotient. divisor divisor Copyright 001
Page 5 of 6 EXPONENT or POWER An exponent signifies the nuber of ties a nuber is to be ultiplied by itself. Exaple: In the expression x, x is the base and is the exponent. ( This eans x is ultiplied by itself ties or x x x x ) To ultiply when the bases are the sae, add the exponents. n n x x x To divide when the bases are the sae, subtract the exponents. n n x x x A negative exponent can be written as its own reciprocal. x 1/ x and x 1 / x Anything (except zero itself) with a 0 exponent equals 1. x 0 1 when x 0 To raise an exponent to a power, ultiply the exponents. n n x x Exponents are distributive through ultiplication and division. ax a x y y A fractional exponent indicates a root. x 1 x since x x x x x 1 1 Copyright 001
Page 6 of 6 EQUATION A stateent of equality between algebraic expressions. Exaple: for the equation x 6, there is a nuber which when substituted for x will resolve the equation to 6 6. That nuber is. The single overall rule to follow in working with equations is that you can do alost anything to an equation as long as the equality is preserved. All of the rules that follow are based on this single rule. Distributive property: (sae as arithetic) ab c ab ac Identical operations on both sides of the equation: Addition: if a b then a c b c Subtraction: if a b then a c b c Multiplication: if a b then a c b c Division: if a b then a / c b / c Siplification: If two ters on the sae side of an equation are identical, except for the algebraic sign (one ter added and the other subtracted), they will cancel each other and can both be eliinated. Additive identity: a b b a b b a 0 a If two ters on the sae side of an equation are identical, one ultiplying and the other dividing that side of the equation, they will cancel each other and can both be eliinated. Multiplicative identity: a b / b a b / b a 1 a Transposition: A ter ay be oved fro one side of the equation to the other by applying the properties stated above. In each of the following equations, the value of x is solved for by isolating it on one side of the equation. Addition: if x a b then x b a Subtraction: if x a b then x b a Multiplication: if x / a b then x b a Division: if x a b then x b / a Copyright 001