Basic Algebra Review

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Algebra Review Exponents (continued) Polynomials (continued) Factoring (continued) Rational Expressions Rational Expressions Equations of Lines Quotient Rules FOIL Expansion for Multiplying Two To find the value(s) for which a rational (continued) TwoVariables (continued) Factoring Trinomials, Z Binomials expression is undefined, set the denominator If a 0, Leading Term Z x2 Intercepts 0 equal to 0 and solve the resulting equation. SIMPLIFYING COMPLEX FRACTIONS i. Zero exponent: a = 1 i. Multiply the first terms. 2 + + Z To find the x-intercept, let y = 0 . To factor ax bx c, a 1 : Lowest Terms Numbers Linear Equations ii. Multiply the outer terms. Method 1 To find the y-intercept, let x = 0 . -n = 1 By Grouping To write a rational expression in lowest terms: ii. Negative exponents: a n iii. Multiply the inner terms. i. Simplify the numerator and denominator FRACTIONS Definition of Subtraction Properties a i. Find m and n such that i. Factor the numerator and denominator. Slope iv. Multiply the last terms. = + = separately. m mn ac and m n b. Suppose (x , y ) and (x , y ) are two differ- Addition and Subtraction x - y = x + -y i. Addition: The same quantity may be a - ii. Divide out common factors. ii. Divide by multiplying the simplified 1 1 2 2 1 2 iii. Quotient rule: = m n v. Collect like terms. Z added to (or subtracted from) each side of n a ii. Then numerator by the reciprocal of the ent points on a line. If x1 x2 , then the i. To add or subtract fractions with the same Subtracting Real Numbers a 2 2 an equality without changing the solution. ax bx c ax mx nx c. OPERATIONS ON RATIONAL simplified denominator. slope is denominator, add or subtract the numera- i. Change the subtraction symbol to the SPECIAL PRODUCTS - tors and keep the same denominator. ii. Multiplication: Each side of an equality iv. Negative to positive: iii. Group the first two terms and the last EXPRESSIONS rise y2 y1 addition symbol. Square of a Binomial two terms. Method 2 m = = . may be multiplied (or divided) by the -m n run x - x ii. To add or subtract fractions with differ- ii. Change the sign of the number being a b 2 2 2 Multiplying Rational Expressions i. Multiply the numerator and denomina- 2 1 same nonzero number without chang- = , a Z 0, b Z 0 x + y = x + 2xy + y iv. Follow the steps for factoring by grouping. ent denominators, find the LCD and -n m 1 2 tor of the complex fraction by the LCD subtracted. ing the solution. b a - 2 = 2 - + 2 i. Multiply numerators and multiply The slope of a vertical line is undefined. write each fraction with this LCD. Then - x y x 2xy y By Trial and Error of all the denominators in the complex iii. Add using the rules for adding real m m 1 2 denominators. follow the procedure in step i. Solving Linear Equalities a b fraction. The slope of a horizontal line is 0. numbers. = , a Z 0, b Z 0 i. Factor a as pq and c as mn. ii. Factor numerators and denominators. a b a b Parallel lines have the same slope. Multiplication i. Simplify each side separately. b a Product of the Sum and Difference of ii. For each such factorization, form the ii. Write in lowest terms. Multiplying Real Numbers Two Terms + + iii. Write expression in lowest terms. Multiply numerators and multiply ii. Isolate the variable term on one side. Scientific Notation product px m qx n and Perpendicular lines have slopes that are 2 2 1 21 2 i. Multiply the absolute value of the two x + y x - y = x - y expand using FOIL. Dividing Rational Expressions SOLVING EQUATIONS WITH negative reciprocals of each other. denominators. iii. Isolate the variable. A number written in scientific notation is in 1 21 2 numbers. * n i. Multiply the first rational expression by RATIONAL EXPRESSIONS Division the form a 10 , where a has one digit in iii. Stop when the expansion matches the EQUATIONS OF LINES ii. If the two numbers have the same sign, front of the decimal point and that digit is Dividing a Polynomial by a original trinomial. the reciprocal of the second rational i. Find the LCD of all denominators in the Multiply the first fraction by the reciprocal APPLICATIONS the product is positve. If the two num- nonzero. To write a number in scientific Monomial expression. equation. Slope–intercept form: y = mx + b, of the second fraction. bers have different signs, the product is i. Assign a variable to the unknown Remainder Theorem notation, move the decimal point to follow Divide each term of the polynomial by the ii. Multiply numerators and multiply where m is the slope, and 0, b is the negative. quantity in the problem. If the polynomial P(x) is divided by x – a, then ii. Multiply each side of the equation by 1 2 the first nonzero digit. If the decimal point monomial: denominators. the LCD. y-intercept. ORDER OF OPERATIONS ii. Write an equation involving the unknown. has been moved n places to the left, the the remainder is equal to P(a). x = # 1 Z + iii. Factor numerators and denominators. Simplify within parentheses, brackets, or Definition of Division: x , y 0 exponent on 10 is n. If the decimal point p q p q iii. Solve the resulting equation. x y y y iii. Solve the equation. = + Factor Theorem Intercept form: + = 1, absolute value bars or above and below has been moved n places to the right, the r r r iv. Write expression in lowest terms. iv. Check that the resulting solutions satisfy a b Division by 0 is undefined. For a polynomial P(x) and number a, fraction bars first, in the following order. FORMULAS exponent on 10 is –n. Finding the Least Common the original equation. Dividing a Polynomial by a if P(a) = 0, then x – a is a factor of P(x). where a, 0 is the x-intercept, and 0, b is i. Apply all exponents. Dividing Real Numbers Denominator (LCD) 1 2 1 2 i. To find the value of one of the variables in Polynomial the y-intercept. i. Divide the absolute value of a formula, given values for the others, sub- SPECIAL FACTORIZATIONS ii. Perform any multiplications or divisions i. Factor each denominator into prime Equations of Lines - = - the numbers. stitute the known values into the formula. Polynomials Use long division or synthetic division. Point–slope form: y y1 m x x1 , from left to right. Difference of Squares factors. TwoVariables 1 2 iii. Perform any additions or subtractions ii. If the signs are the same, the answer is ii. To solve a formula for one of the vari- A polynomial is an algebraic expression 2 2 where m is the slope and x1, y1 is any Graphing Simple Polynomials x - y = x + y x - y ii. List each different factor the greatest 1 2 from left to right. positive. If the signs are different, the ables, isolate that variable by treating made up of a term or a finite sum of terms 1 21 2 number of times it appears in any one An ordered pair is a solution of an equation point on the line. answer is negative. with real or complex coefficients and whole the other variables as constants (num- i. Determine several points (ordered pairs) Perfect Square Trinomials denominator. if it satisfies the equation. Standard form: Ax + By = C bers) and using the steps for solving number exponents. satisfying the polynomial equation. 2 + + 2 = + 2 VARIABLES, EXPRESSIONS, x 2xy y x y If the value of either variable in an equation PROPERTIES OF REAL NUMBERS equations. 1 2 iii. Multiply the factors from step ii. Vertical line: x = a AND EQUATIONS The degree of a term is the sum of the ii. Plot the points. x2 - 2xy + y2 = x - y 2 is given, the value of the other variable can Commutative Properties exponents on the variables. The degree of a 1 2 Writing a Rational Expression with a be found by substitution. Horizontal line: y = b An expression containing a variable is evalu- iii. Connect the points with a smooth Difference of Cubes ated by substituting a given number for the + = + polynomial is the highest degree amongst all of curve. Specified Denominator a b b a Exponents 3 - 3 = - 2 + + 2 its terms. x y x y x xy y i. Factor both denominators. GRAPHING LINEAR EQUATIONS variable. ab = ba 1 21 2 Values for a variable that make an equation For any integers m and n, the following rules A monomial is a polynomial with only one Sum of Cubes ii. Determine what factors the given To graph a linear equation: Systems of Linear Equations hold: term. Factoring true are solutions of the equation. Associative Properties x3 + y3 = x + y x2 - xy + y2 denominator must be multiplied by to 1 21 2 i. Find at least two ordered pairs that satisfy TWO VARIABLES a + b + c = a + b + c Product Rule A binomial is a polynomial with exactly two Finding the Greatest Common Factor equal the one given. the equation. REAL NUMBERS AND 1 2 1 2 m # n m+n terms. An ordered pair is a solution of a system if it ab c = a bc a a = a (GCF) SOLVING QUADRATIC EQUATIONS iii. Multiply the rational expression by that ii. Plot the corresponding points. (An THE NUMBER LINE 1 2 1 2 A trinomial is a polynomial with exactly BY FACTORING factor divided by itself. satisfies all the equations at the same time. Power Rules i. Include the largest numerical factor of ordered pair (a, b) is plotted by starting at a is less than b if a is to the left of b on the Distributive Properties three terms. the origin, moving a units along the x-axis Graphing Method i. am n = amn each term. Zero-Factor Property Adding or Subtracting Rational number line. a b + c = ab + ac 1 2 = = = Expressions and then b units along the y-axis.) i. Graph each equation of the system on 1 2 m = m m OPERATIONS ON POLYNOMIALS ii. Include each variable that is a factor of If ab 0, then a 0 or b 0. The additive inverse of x is –x. b + c a = ba + ca ii. ab a b every term raised to the smallest expo- iii. Draw a straight line through the points. the same axes. 1 2 1 2 Solving Quadratic Equations i. Find the LCD. The absolute value of x, denoted |x|, is the m m Adding Polynomials nent that appears in a term. a a ii. Rewrite each rational expression with Special Graphs ii. Find the coordinates of the point of Identity Properties = Z Add like terms. i. Write in standard form: distance (a positive number) between x and iii. m, b 0 = intersection. + = + = a b b b Factoring by Grouping 2 + + = the LCD as denominator. x a is a vertical line through the point 0 on the number line. a 0 a 0 a a Subtracting Polynomials ax bx c 0 iii. Verify that the point satisfies all the # # more➤ i. Group the terms. iii. If adding, add the numerators to get the a, 0 . a 1 = a 1 a = a ii. Factor. 1 2 equations. OPERATIONS ON REAL NUMBERS Change the sign of the terms in the second ii. Factor out the greatest common factor numerator of the sum. If subtracting, y = b is a horizontal line through the point Inverse Properties polynomial and add to the first polynomial. in each group. iii. Use the zero-factor property to set each subtract the second numerator from the a, b . Substitution Method Adding Real Numbers ISBN-13: 978-0-321-39473-6 factor to zero. first numerator to get the difference. The 1 2 + - = - + = Multiplying Polynomials iii. Factor a common binomial factor from The graph of Ax + By = 0 goes through i. Solve one equation for either variable. To add two numbers with the same sign, add a a 0 a a 0 ISBN-10: 0-321-39473-9 LCD is the denominator of the sum. 1 2 1 2 i. Multiply each term of the first polyno- the result of step ii. iv. Solve each resulting equation to find the origin. Find and plot another point that ii. Substitute that variable into the other their absolute values. The sum has the same # 1 1 # 90000 each solution. iv. Write expression in lowest terms. satisfies the equation, and then draw the line a = 1 a = 1 a Z 0 mial by each term of the second poly- iv. Try various groupings, if necessary. equation. sign as each of the numbers being added. 1 2 more➤ through the two points. a a nomial. iii. Solve the equation from step ii. To add two numbers with different signs, Factoring Trinomials, more➤ subtract their absolute values. The sum has Simplifying Algebraic Expressions ii. Collect like terms. Leading Term x2 iv. Substitute the result from step iii into the more➤ 2 equation from step i to find the remain- the sign of the number with the larger abso- When adding or subtracting algebraic expres- To factor x + bx + c, a Z 1: lute value. sions, only like terms can be combined. ing value. i. Find m and n such that more➤ 9 780321 394736 mn = c and m + n = b. ii. Then x2 + bx + c = x + m x + n . 1 21 2 1 iii. Verify by using FOIL2 expansion. 3 more➤ SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 1

Algebra Review

Systems of Linear Equations Inequalities and Absolute Inequalities and Absolute Roots and Radicals Roots and Radicals Quadratic Equations, Inverse, Exponential, and Conic Sections Sequences and Series (continued) Value: One Variable Value: One Variable Radical Expressions and Graphs (continued) Inequalities, and Functions Logarithmic Functions and Nonlinear Systems A sequence is a list of terms t1, t2, t3, … (continued) n (continued) (continued) (continued) Elimination Method Properties 2a = b means bn = a. (finite or infinite) whose general (nth) term is COMPLEX NUMBERS denoted t . i. Write the equations in standard form: i. Addition: The same quantity may be Graphing a Linear Inequality n Number and Exponential Functions n 2a is the principal or positive nth root of a. = - 2 =- ELLIPSE + = added to (or subtracted from) each side The imaginary unit is i 2 1 , so i 1 . Discriminant Type of Solution 7 Z = x A series is the sum of the terms in a sequence. Ax By C. i. If the inequality sign is replaced by an 0 For a 0, a 1, f x a defines the of an inequality without changing the - 2a is the negative nth root of a. b2 - 4ac 7 0 Two real solutions 1 2 Equation of an Ellipse (Standard equals sign, the resulting line is the For b 7 0 , 2-b = i2b . To multiply rad- exponential function with base a. ii. Multiply one or both equations by appro- solution. 2 ARITHMETIC SEQUENCES equation of the boundary. n n = b - 4ac = 0 One real solution = x Position, Major Axis along x-axis) priate numbers so that the sum of the ii. Multiplication by positive numbers: 2a |a| if n is even. icals with negative radicands, first change Properties of the graph of f x a : ii. Draw the graph of the boundary line, 2 - 6 1 2 2 2 An arithmetic sequence is a sequence in coefficient of one variable is 0. Each side of an inequality may be multi- n b 4ac 0 Two complex solutions x y making the line solid if the inequality 2an = a if n is odd. each factor to the form i2b . i. Contains the point (0, 1) + = 1, a 7 b 7 0 which the difference between successive plied (or divided) by the same positive 2 2 iii. Add the equations to eliminate one of involves … or Ú or dashed if the + ii. If a 7 1 , the graph rises from left to a b terms is a constant. number without changing the solution. Rational Exponents A complex number has the form a bi, QUADRATIC FUNCTIONS the variables. inequality involves < or >. right. If 0 6 a 6 1 , the graph falls 1 n n 1 n n where a and b are real numbers. Let a1 be the first term, an be the nth term, iv. Solve the equation that results from step iii. Multiplication by negative numbers: a > : If 1a is real, then a > = 1a . Standard Form from left to right. is the equation of an ellipse centered at the iii. Choose any point not on the line as a and d be the common difference. iii. If each side of an inequality is multi- m n = 2 + + origin, whose x-intercepts (vertices) are test point and substitute its coordinates a > : If m and n are positive integers with OPERATIONS ON COMPLEX f x ax bx c, for a, b, c real, iii. The x-axis is an asymptote. = plied (or divided) by the same negative 1 2 - Common difference: d an+1 – an v. Substitute the solution from step iv into into the inequality. 1 n a Z 0. - q q q a, 0 and a, 0 and y-intercepts are number, the direction of the inequality m/n in lowest terms and a > is real, then NUMBERS iv. Domain: ( , ); Range: (0, ) 1 2 1 2 either of the original equations to find The graph is a parabola, opening up if 0, b 0, -b . Foci are c, 0 and -c, 0 = + - symbol is reversed. iv. If the test point satisfies the inequality, m n 1 n m Adding and Subtracting Complex , nth term: an a1 n 1 d the value of the remaining variable. a > = a > . a 7 0, down if a 6 0 . The vertex is Logarithmic Functions 1 2 1 2 1 2 1 2 1 2 shade the region that includes the test = 2 2 - 2 Solving Linear Inequalities 1 2 Numbers The logarithmic function is the inverse of the where c a b . Sum of the first n terms: Notes: If the result of step iii is a false state- point; otherwise, shade the region that 1 n m n - - 2 If a > is not real, then a > is not real. Add (or subtract) the real parts and add (or b 4ac b exponential function: ment, the graphs are parallel lines and there i. Simplify each side separately. does not include the test point. 2a , 4a . Equation of an Ellipse (Standard n n a b y = + = + - is no solution. subtract) the imaginary parts. y = log x meansx = a . Position, Major Axis along y-axis) Sn a1 an 2a1 n 1 d ii. Isolate the variable term on one side. SIMPLIFYING RADICAL EXPRESSIONS = - b a 21 2 23 1 2 4 If the result of step iii is a true statement, such n n Multiplying Complex Numbers The axis of symmetry is x 2a . 7 Z = 2 2 iii. Isolate the variable. (Reverse the Product Rule: If 1a and 1b are real and n For a 0, a 1, g x loga x defines y x as 0 = 0, the graphs are the same line, and the Functions 1 2 + = 7 7 inequality symbol when multiplying or is a natural number, then Multiply using FOIL expansion and using the logarithmic function with base a. 2 2 1, a b 0 GEOMETRIC SEQUENCES solution is every ordered pair on either line (of 2 =- Vertex Form a b dividing by a negative number.) n # n n i 1 to reduce the result. = which there are infinitely many). Function Notation 1a 1b = 1ab. f x = a x - h 2 + k. The vertex is Properties of the graph of g x loga x : A geometric sequence is a sequence in which Dividing Complex Numbers 1 2 1 2 1 2 is the equation of an ellipse centered at the Solving Compound Inequalities A function is a set of ordered pairs (x, y) such n n = i. Contains the points (1, 0) and (a, 1) the ratio of successive terms is a constant. Quotient Rule: If 1a and 1b are real and h, k . The axis of symmetry is x h. THREE VARIABLES i. Solve each inequality in the compound, that for each first component x, there is one and Multiply the numerator and the denominator 1 2 7 origin, whose x-intercepts (vertices) are Let t be the first term, t be the nth term, n is a natural number, then Horizontal Parabola ii. If a 1 , the graph rises from left to - 1 n inequality individually. only one second component y. The set of first by the conjugate of the denominator. right. If 0 6 a 6 1 , the graph falls b, 0 and b, 0 and y-intercepts are and r be the common ratio. i. Use the elimination method to eliminate n = 2 + + 1 2 -1 2 - components is called the domain, and the set n a 1a The graph of x ay by c, is a from left to right. 0, a 0, a . Foci are 0, c and 0, c , any variable from any two of the original ii. If the inequalities are joined with and, = 1 2 1 2 1 2 1 2 t + of second components is called the range. n . horizontal parabola, opening to the right if = 2 2 n 1 equations. then the solution set is the intersection A b 1 iii. The y-axis is an asymptote. where c 2a - b . Common ratio: r = = b Quadratic Equations, a 7 0, to the left if a 6 0 . Note that this is t ii. Eliminate the same variable from any of the two individual solution sets. y f(x) defines y as a function of x. q - q q n Inequalities, and Functions not the graph of a function. iv. Domain: (0, ); Range: ( , ) = n-1 other two equations. iii. If the inequalities are joined with or, To write an equation that defines y as a HYPERBOLA nth term: tn t1r OPERATIONS ON Logarithm Rules then the solution set is the union of the function of x in function notation, solve the Equation of a Hyperbola (Standard iii. Steps i and ii produce a system of two RADICAL EXPRESSIONS SOLVING QUADRATIC EQUATIONS QUADRATIC INEQUALITIES = + equations in two variables. Use the elimi- two individual solution sets. equation for y and replace y by f (x). Product rule: log a xy log a x log a y Position, Opening Left and Right) Sum of the first n terms: Adding and Subtracting: Only radical Solving Quadratic (or Higher- x = - n nation method for two-variable systems to To evaluate a function written in function Square Root Property Quotient rule: log log x log y - Solving Absolute Value Equations expressions with the same index and the a y a a 2 y2 t1 r 1 solve for the two variables. notation for a given value of x, substitute the Degree Polynomial) Inequalities r x S = 1 2, r Z 1 and Inequalities If a is a complex number, then the solutions = - = n same radicand can be combined. 2 Power rule: log a x rlog a x 2 2 1 r - 1 value wherever x appears. to = are = 1 and =-1 . i. Replace the inequality sign by an a b iv. Substitute the values from step iii into x a x a x a log x x Suppose k is positive. Multiplying: Multiply binomial radical Special properties: a a = x , log a = x any of the original equations to find the equality sign and find the real-valued a ƒ + ƒ = Variation expressions by using FOIL expansion. Solving Quadratic Equations by is the equation of a hyperbola centered at Sum of the terms of an infinite geometric value of the remaining variable. To solve ax b k , solve the solutions to the equation. Change-of-base rule: For a 7 0 , a Z 1 , If there exists some real number (constant) k Completing the Square t1 compound equation Dividing: Rationalize the denominator by ii. Use the solutions from step i to divide the origin, whose x-intercepts (vertices) are sequence with |r| < 1: S = + = + =- such that: To solve ax2 + bx + c = 0, a Z 0 : logb x - ax b k or ax b k. multiplying both the numerator and denomi- the real number line into intervals. 7 Z 7 = 1 r APPLICATIONS n n b 0, b 1, x 0, log a x . a, 0 and -a, 0 . Foci are c, 0 and y = kx , then y varies directly as x . nator by the same expression. If the denomi- Z log a i. If a 1 , divide each side by a . iii. Substitute a test number from each interval b 1 2 1 2 1 2 i. Assign variables to the unknown To solve ƒ ax + b ƒ 7 k , solve the nator involves the sum of an integer and a -c, 0 , where c = 2a2 + b2 . k ii. Write the equation with the variable into the original inequality to determine the Exponential, Logarithmic Equations 1 2 quantities in the problem. compound inequality = n square root, the expression used will be b The Binomial Theorem y n, then y varies inversely as x . terms on one side of the equals sign and intervals that belong to the solution set. 7 Z Asymptotes are y =; x . ii. Write a system of equations that relates ax + b 7 k orax + b 6-k . x chosen to create a difference of squares. Suppose b 0, b 1. a the constant on the other. iv. Consider the endpoints separately. x = y = Equation of a Hyperbola (Standard the unknowns. y = kxz, then y varies jointly as x and z. Solving Equations Involving Radicals i. If b b , then x y. Factorials To solve ƒ ax + b ƒ 6 k , solve the com- iii. Take half the coefficient of x and square Position, Opening Up and Down) iii. Solve the system. 7 7 = For any positive integer n, pound inequality Operations on Functions i. Isolate one radical on one side of the it. Add the square to each side of the ii. If x 0, y 0, then log b x log b y is equivalent to x = y . y2 x2 n! = n n - 1 n - 2 Á 3 2 1 -k 6 ax + b 6 k. If f(x) and g(x) are functions, then the equation. equation. Inverse, Exponential, and - = 1 1 21 2 1 21 21 2 MATRIX ROW OPERATIONS = y = 2 2 and following functions are derived from f and g: ii. Raise both sides of the equation to a iv. Factor the perfect square trinomial and iii. If log b x y, then b x. a b i. Any two rows of the matrix may be Logarithmic Functions 0! = 1. To solve an absolute value equation of the f + g x = f x + g x power that equals the index of the write it as the square of a binomial. interchanged. form ƒ ax + b ƒ = ƒ cx + d ƒ , solve the is the equation of a hyperbola centered at the 1 21 2 1 2 1 2 radical. Combine the constants on the other side. Inverse Functions Conic Sections Binomial Coefficient ii. All the elements in any row may be compound equation f - g x = f x - g x origin, whose y-intercepts (vertices) are 0, a 1 21 2 1 2 1 2 iii. Solve the resulting equation; if it still v. Use the square root property to determine If any horizontal line intersects the graph of 1 2 multiplied by any nonzero real number. + = + and Nonlinear Systems For any nonnegative integers n and r, with ax b cx d or = # contains a radical, repeat steps i and ii. the solutions. a function in, at most, one point, then the and 0, -a . Foci are 0, c and 0, -c , fg x f x g x 1 2 1 2 1 2 … n = = n! iii. Any row may be modified by adding to + =- + 1 21 2 1 2 1 2 function is one to one and has an inverse. 2 2 r n, nCp . ax b cx d . iv. The resulting solutions are only candi- Quadratic Formula CIRCLE where c = 2a + b . Asymptotes are a r b r! n - r ! the elements of the row the product of 1 2 f f x 1 2 more➤ = 1 2 Z dates. Check which ones satisfy the orig- 2 + + = If y = f ( x) is one to one, then the equation =;a a real number and the elements of (x) , g x 0 The solutions of ax bx c 0 , –1 Equation of a Circle: Center-Radius y x. a g b g x 1 2 inal equation. Candidates that do not Z that defines the inverse function f is found b n another row. 1 2 a 0 are given by - 2 + - 2 = 2 The binomial expansion of (x y) has n + 1 check are extraneous (not part of the by interchanging x and y, solving for y, and x h y k r A system of equations can be represented by Composition of f and g: solution set). - 2 - 1 2 1 2 SOLVING NONLINEAR SYSTEMS b 2b 4ac –1 terms. The (r 1)st term of the binomial more➤ = . replacing y with f (x). is the equation of a circle with radius r and a matrix and solved by matrix methods. f g x = f g x x A nonlinear system contains multivariable 1 21 2 3 1 24 2a The graph of f –1 is the mirror image of the center at h, k . Write an augmented matrix and use row 1 2 terms whose degrees are greater than one. expansion of (x y)n for r 0, 1, …, n is operations to reduce the matrix to row b2 - 4ac is called the discriminant and graph of f with respect to the line y = x . Equation of a Circle: General more➤ A nonlinear system can be solved by the echelon form. determines the number and type of solutions. x2 + y2 + ax + by + c = 0 substitution method, the elimination n! n-r r more➤ x y . method, or a combination of the two. r! n - r ! Given an equation of a circle in general 1 2 form, complete the squares on the x and y terms separately to put the equation into 4 5 center-radius form. 6 more➤ SC_03213974739_rp08.qxd 1/15/08 9:15 AM Page 2

Algebra Review