# Definition of Subtraction x - y = x + 1-y2. Subtracting Real Numbers

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2 Exponents Quotient Rules If a Z 0, i Zero exponent: a 0 = 1 ii Negative exponents: a -n = 1 a n iii Quotient rule: iv Negative to positive: a -m b -n = bn a m, a Z 0, b Z 0 a a -m b b = a b m, a b a Z 0, b Z 0 Scientific Notation A number written in scientific notation is in the form a * 10 n, where a has one digit in front of the decimal point and that digit is nonzero To write a number in scientific notation, move the decimal point to follow the first nonzero digit If the decimal point has been moved n places to the left, the exponent on 10 is n If the decimal point has been moved n places to the right, the exponent on 10 is n Polynomials A polynomial is an algebraic expression made up of a term or a finite sum of terms with real or complex coefficients and whole number exponents The degree of a term is the sum of the exponents on the variables The degree of a polynomial is the highest degree amongst all of its terms A monomial is a polynomial with only one term A binomial is a polynomial with exactly two terms A trinomial is a polynomial with exactly three terms OPERATIONS ON POLYNOMIALS Adding Polynomials Add like terms a m a n = am - n Subtracting Polynomials Change the sign of the terms in the second polynomial and add to the first polynomial Multiplying Polynomials i Multiply each term of the first polynomial by each term of the second polynomial ii Collect like terms Polynomials FOIL Expansion for Multiplying Two Binomials i Multiply the first terms ii Multiply the outer terms iii Multiply the inner terms iv Multiply the last terms v Collect like terms SPECIAL PRODUCTS Square of a Binomial 1 x + y2 2 = x 2 + 2xy + y 2 1 x - y2 2 = x 2-2xy + y 2 Product of the Sum and Difference of Two Terms 1x + y21 x - y2 = x 2 - y 2 Dividing a Polynomial by a Monomial Divide each term of the polynomial by the monomial: p + q = p r r + q r Dividing a Polynomial by a Polynomial Use long division or synthetic division Graphing Simple Polynomials i Determine several points (ordered pairs) satisfying the polynomial equation ii Plot the points iii Connect the points with a smooth curve Factoring Finding the Greatest Common Factor (GCF) i Include the largest numerical factor of each term ii Include each variable that is a factor of every term raised to the smallest exponent that appears in a term Factoring by Grouping i Group the terms ii Factor out the greatest common factor in each group iii Factor a common binomial factor from the result of step ii iv Try various groupings, if necessary Factoring Trinomials, Leading Term x 2 To factor x 2 + bx + c, a Z 1: i Find m and n such that mn = c and m + n = b ii Then x 2 + bx + c = 1x + m21x + n2 iii Verify by using FOIL 2 expansion Factoring Factoring Trinomials, Leading Term Z x 2 To factor ax 2 + bx + c, a Z 1: By Grouping i Find m and n such that mn = ac and m + n = b ii Then ax 2 bx c ax 2 mx nx c iii Group the first two terms and the last two terms iv Follow the steps for factoring by grouping By Trial and Error i Factor a as pq and c as mn ii For each such factorization, form the product 1 px + m21qx + n2 and expand using FOIL iii Stop when the expansion matches the original trinomial Remainder Theorem If the polynomial P(x) is divided by x a, then the remainder is equal to P(a) Factor Theorem For a polynomial P(x) and number a, if P(a) = 0, then x a is a factor of P(x) SPECIAL FACTORIZATIONS Difference of Squares x 2 - y 2 = 1x + y21x - y2 Perfect Square Trinomials x 2 + 2xy + y 2 = 1x + y2 2 x 2-2xy + y 2 = 1x - y2 2 Difference of Cubes x 3 - y 3 = 1x - y21x 2 + xy + y 2 2 Sum of Cubes x 3 + y 3 = 1x + y21x 2 - xy + y 2 2 SOLVING QUADRATIC EQUATIONS BY FACTORING Zero-Factor Property If ab = 0, then a = 0 or b = 0 Solving Quadratic Equations i Write in standard form: ax 2 + bx + c = 0 ii Factor iii Use the zero-factor property to set each factor to zero iv Solve each resulting equation to find each solution

3 Rational Expressions To find the value(s) for which a rational expression is undefined, set the denominator equal to 0 and solve the resulting equation Lowest Terms To write a rational expression in lowest terms: i Factor the numerator and denominator ii Divide out common factors OPERATIONS ON RATIONAL EXPRESSIONS Multiplying Rational Expressions i Multiply numerators and multiply denominators ii Factor numerators and denominators iii Write expression in lowest terms Dividing Rational Expressions i Multiply the first rational expression by the reciprocal of the second rational expression ii Multiply numerators and multiply denominators iii Factor numerators and denominators iv Write expression in lowest terms Finding the Least Common Denominator (LCD) i Factor each denominator into prime factors ii List each different factor the greatest number of times it appears in any one denominator iii Multiply the factors from step ii Writing a Rational Expression with a Specified Denominator i Factor both denominators ii Determine what factors the given denominator must be multiplied by to equal the one given iii Multiply the rational expression by that factor divided by itself Adding or Subtracting Rational Expressions i Find the LCD ii Rewrite each rational expression with the LCD as denominator iii If adding, add the numerators to get the numerator of the sum If subtracting, subtract the second numerator from the first numerator to get the difference The LCD is the denominator of the sum iv Write expression in lowest terms Rational Expressions SIMPLIFYING COMPLEX FRACTIONS Method 1 i Simplify the numerator and denominator separately ii Divide by multiplying the simplified numerator by the reciprocal of the simplified denominator Method 2 i Multiply the numerator and denominator of the complex fraction by the LCD of all the denominators in the complex fraction ii Write in lowest terms SOLVING EQUATIONS WITH RATIONAL EXPRESSIONS i Find the LCD of all denominators in the equation ii Multiply each side of the equation by the LCD iii Solve the resulting equation iv Check that the resulting solutions satisfy the original equation Equations of Lines Two Variables An ordered pair is a solution of an equation if it satisfies the equation If the value of either variable in an equation is given, the value of the other variable can be found by substitution GRAPHING LINEAR EQUATIONS To graph a linear equation: i Find at least two ordered pairs that satisfy the equation ii Plot the corresponding points (An ordered pair (a, b) is plotted by starting at the origin, moving a units along the x-axis and then b units along the y-axis) iii Draw a straight line through the points Special Graphs x = a is a vertical line through the point 1a, 02 y = b is a horizontal line through the point 1a, b2 The graph of Ax + By = 0 goes through the origin Find and plot another point that satisfies the equation, and then draw the line through the two points Equations of Lines Two Variables Intercepts To find the x-intercept, let y = 0 To find the y-intercept, let x = 0 Slope Suppose ( x 1, y 1 ) and ( x 2, y 2 ) are two different points on a line If x 1 Z x 2, then the slope is m = rise run = y 2 - y 1 x 2 - x 1 The slope of a vertical line is undefined The slope of a horizontal line is 0 Parallel lines have the same slope Perpendicular lines have slopes that are negative reciprocals of each other EQUATIONS OF LINES Slope intercept form: y = mx + b, where m is the slope, and 10, b2 is the y-intercept x Intercept form:, a + y b = 1 where 1a, 02 is the x-intercept, and 10, b2 is the y-intercept Point slope form: y - y 1 = m1x - x 1 2, where m is the slope and 1x 1, y 1 2 is any point on the line Standard form: Ax + By = C Vertical line: x = a Horizontal line: y = b Systems of Linear Equations TWO VARIABLES An ordered pair is a solution of a system if it satisfies all the equations at the same time Graphing Method i Graph each equation of the system on the same axes ii Find the coordinates of the point of intersection iii Verify that the point satisfies all the equations Substitution Method i Solve one equation for either variable ii Substitute that variable into the other equation iii Solve the equation from step ii iv Substitute the result from step iii into the equation from step i to find the remaining value 3

4 Algebra Review Systems of Linear Equations Elimination Method i Write the equations in standard form: Ax + By = C ii Multiply one or both equations by appropriate numbers so that the sum of the coefficient of one variable is 0 iii Add the equations to eliminate one of the variables iv Solve the equation that results from step iii v Substitute the solution from step iv into either of the original equations to find the value of the remaining variable Notes: If the result of step iii is a false statement, the graphs are parallel lines and there is no solution If the result of step iii is a true statement, such as 0 = 0, the graphs are the same line, and the solution is every ordered pair on either line (of which there are infinitely many) THREE VARIABLES i Use the elimination method to eliminate any variable from any two of the original equations ii Eliminate the same variable from any other two equations iii Steps i and ii produce a system of two equations in two variables Use the elimination method for two-variable systems to solve for the two variables iv Substitute the values from step iii into any of the original equations to find the value of the remaining variable APPLICATIONS i Assign variables to the unknown quantities in the problem ii Write a system of equations that relates the unknowns iii Solve the system MATRIX ROW OPERATIONS i Any two rows of the matrix may be interchanged ii All the elements in any row may be multiplied by any nonzero real number iii Any row may be modified by adding to the elements of the row the product of a real number and the elements of another row A system of equations can be represented by a matrix and solved by matrix methods Write an augmented matrix and use row operations to reduce the matrix to row echelon form Inequalities and Absolute Value: One Variable Properties i Addition: The same quantity may be added to (or subtracted from) each side of an inequality without changing the solution ii Multiplication by positive numbers: Each side of an inequality may be multiplied (or divided) by the same positive number without changing the solution iii Multiplication by negative numbers: If each side of an inequality is multiplied (or divided) by the same negative number, the direction of the inequality symbol is reversed Solving Linear Inequalities i Simplify each side separately ii Isolate the variable term on one side iii Isolate the variable (Reverse the inequality symbol when multiplying or dividing by a negative number) Solving Compound Inequalities i Solve each inequality in the compound, inequality individually ii If the inequalities are joined with and, then the solution set is the intersection of the two individual solution sets iii If the inequalities are joined with or, then the solution set is the union of the two individual solution sets Solving Absolute Value Equations and Inequalities Suppose k is positive To solve ƒ ax + b ƒ = k, solve the compound equation ax + b = k or ax + b = -k To solve ƒ ax + b ƒ 7 k, solve the compound inequality ax + b 7 k or ax + b 6 -k To solve ƒ ax + b ƒ 6 k, solve the compound inequality -k 6 ax + b 6 k To solve an absolute value equation of the form ƒ ax + b ƒ = ƒ cx + d ƒ, solve the compound equation ax + b = cx + d or ax + b = -1cx + d2 Inequalities and Absolute Value: One Variable Graphing a Linear Inequality i If the inequality sign is replaced by an equals sign, the resulting line is the equation of the boundary ii Draw the graph of the boundary line, making the line solid if the inequality involves or Ú or dashed if the inequality involves < or > iii Choose any point not on the line as a test point and substitute its coordinates into the inequality iv If the test point satisfies the inequality, shade the region that includes the test point; otherwise, shade the region that does not include the test point Functions Function Notation A function is a set of ordered pairs (x, y) such that for each first component x, there is one and only one second component y The set of first components is called the domain, and the set of second components is called the range y = f(x) defines y as a function of x To write an equation that defines y as a function of x in function notation, solve the equation for y and replace y by f (x) To evaluate a function written in function notation for a given value of x, substitute the value wherever x appears Variation If there exists some real number (constant) k such that: y = kx n, then y varies directly as x n y = k, then y varies inversely as x n x n y = kxz, then y varies jointly as x and z Operations on Functions If f(x) and g(x) are functions, then the following functions are derived from f and g: 1f + g21x2 = f1x2 + g1x2 1f - g21x2 = f1x2 - g1x2 1fg21x2 = f1x2 # g1x2 a f f 1x2 b(x) = g g1x2, g1x2 Z 0 Composition of f and g: 1f g21x2 = f 3g1x24 4

6 Inverse, Exponential, and Logarithmic Functions Exponential Functions For a 7 0, a Z 1, f1x2 = a x defines the exponential function with base a Properties of the graph of f1x2 = a x : i Contains the point (0, 1) ii If a 7 1, the graph rises from left to right If 0 6 a 6 1, the graph falls from left to right iii The x-axis is an asymptote iv Domain: (- q, q) ; Range: (0, q) Logarithmic Functions The logarithmic function is the inverse of the exponential function: y = log means x = a y a x For a 7 0, a Z 1, g1x2 = log a x defines the logarithmic function with base a Properties of the graph of g1x2 = log a x: i Contains the points (1, 0) and (a, 1) ii If a 7 1, the graph rises from left to right If 0 6 a 6 1, the graph falls from left to right iii The y-axis is an asymptote iv Domain: (0, q) ; Range: (- q, q) Logarithm Rules Product rule: log a xy = log Quotient rule: log a x a x + log a y Power rule: log a x r y = log a x - log a y = r log a x Special properties: a log a x = x, log a a x = x Change-of-base rule: For a 7 0, a Z 1,,,, log a x = log b x b 7 0 b Z 1 x 7 0 log b a Exponential, Logarithmic Equations Suppose b 7 0, b Z 1 i If b x = b y, then x = y ii If x 7 0, y 7 0, then log b x = log b y is equivalent to x = y iii If log b x = y, then b y = x Conic Sections and Nonlinear Systems CIRCLE Equation of a Circle: Center-Radius 1x - h y - k2 2 = r 2 is the equation of a circle with radius r and center at 1h, k2 Equation of a Circle: General x 2 + y 2 + ax + by + c = 0 Given an equation of a circle in general form, complete the squares on the x and y terms separately to put the equation into center-radius form Conic Sections and Nonlinear Systems ELLIPSE Equation of an Ellipse (Standard Position, Major Axis along x-axis) x 2, a 7 b 7 0 a 2 + y2 b 2 = 1 is the equation of an ellipse centered at the origin, whose x-intercepts (vertices) are 1a, 02 and 1-a, 02 and y-intercepts are 10, b2 10, -b2 Foci are 1c, 02 and 1-c, 02, where c = 2a 2 - b 2 Equation of an Ellipse (Standard Position, Major Axis along y-axis) y 2 a 2 + x 2 b 2 = 1, a 7 b 7 0 is the equation of an ellipse centered at the origin, whose x-intercepts (vertices) are 1b, 02 and 1-b, 02 and y-intercepts are 10, a2 10, -a2 Foci are 10, c2 and 10, -c2, where c = 2a 2 - b 2 HYPERBOLA Equation of a Hyperbola (Standard Position, Opening Left and Right) x 2 a 2 - y2 b 2 = 1 is the equation of a hyperbola centered at the origin, whose x-intercepts (vertices) are 1a, 02 and 1-a, 02 Foci are 1c, 02 and 1-c, 02, where c = 2a 2 + b 2 Asymptotes are y = ; b a x Equation of a Hyperbola (Standard Position, Opening Up and Down) y 2 a 2 - x2 b 2 = 1 is the equation of a hyperbola centered at the origin, whose y-intercepts (vertices) are 10, a2 and 10, -a2 Foci are 10, c2 and 10, -c2, where c = 2a 2 + b 2 Asymptotes are y = ; a b x SOLVING NONLINEAR SYSTEMS A nonlinear system contains multivariable terms whose degrees are greater than one A nonlinear system can be solved by the substitution method, the elimination method, or a combination of the two 6 Sequences and Series A sequence is a list of terms t 1, t 2, t 3, (finite or infinite) whose general (nth) term is denoted t n A series is the sum of the terms in a sequence ARITHMETIC SEQUENCES An arithmetic sequence is a sequence in which the difference between successive terms is a constant Let a 1 be the first term, a n be the nth term, and d be the common difference Common difference: d = a n+1 a n nth term: a n = a 1 + 1n - 12d Sum of the first n terms: S n = n 2 1a 1 + a n 2 = n 2 32a 1 + 1n - 12d4 GEOMETRIC SEQUENCES A geometric sequence is a sequence in which the ratio of successive terms is a constant Let t 1 be the first term, t n be the nth term, and r be the common ratio Common ratio: r = t n + 1 t n nth term: t n = t 1 r n - 1 Sum of the first n terms: Sum of the terms of an infinite geometric t 1 sequence with r < 1: S = 1 - r The Binomial Theorem Factorials S n = t 11r n - 12 r - 1, r Z 1 For any positive integer n, n! = n1n - 121n - 22 Á and 0! = 1 Binomial Coefficient For any nonnegative integers n and r, with a n r b = n! r n, nc p = r!1n - r2! The binomial expansion of (x y) n has n + 1 terms The (r 1)st term of the binomial expansion of (x y) n for r 0, 1,, n is n! r!1n - r2! xn - r y r

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