ADVANCED ALGEBRA PROBLEMS
Mr. X helps math students better understand Advanced Algebra. Our sample math problems are designed to provide the necessary practice to know and understand the ideas and principles of advanced algebra. The sample problems reinforce the advanced algebra lessons available to our subscribers. Check out our free samples below, as well as the advanced algebra problem set.
Advanced Algebra Sample Problem 1
Advanced Algebra Sample Problem 2
Advanced Algebra Sample Problems
General Topics
Algebra II 


Title/Subject  Description  
Adding and Subtracting Complex Numbers  Addition and subtraction of complex numbers of the form: (a + bi) or (c  di).  
Advanced Algebra Problem 301  Exercises of Powers and Roots. Twelve problems; I work the first six, you work the last six. 

Advanced Algebra Problem 302  More exercises of powers and roots. Ten problems; I work five and leave five for you. 

Advanced Algebra Problem 303  More exercises of powers and roots. Eight problems; I work four and leave four for you to do. 

Advanced Algebra Problem 304  We solve an inequality with a fraction of linear binomials by finding Critical Points of the function.  
Advanced Algebra Problem 304a  Ten problems with powers and roots; I work three and leave seven problems for you to work on your own. This lesson is appropriate for good students in Basic Algebra. 

Advanced Algebra Problem 305  We graph a linear inequality. Although this is more advanced than "y = mx + b," we still use that familiar linear relation. 

Advanced Algebra Problem 305a  These problems with radicals and roots are appropriate for some students in Basic Algebra. I work five and leave five problems to work yourself. 

Advanced Algebra Problem 306  We graph a linear inequality.  
Advanced Algebra Problem 306a  These problems can be done by advanced students in Basic Algebra. Ten problems; I work four and leave six for you to work. 

Advanced Algebra Problem 309  We solve an inequality in one variable with AND logic and graph the solution set. Basic Algebra students doing well can do this in a firstyear course as well. 

Advanced Algebra Problem 310  We solve an inequality with Absolute Value in one variable. We show a "shortcut," but you need to understand the language of algebra. 

Advanced Algebra Problem 322  We solve (by two different methods) parametric equations that are functions of "t" by expressing them in terms of x and y.  
Advanced Algebra Problem 328  We solve an inequality in one variable with an Absolute Value condition.  
Advanced Algebra Problem 330  We determine the domains of three functions. This problem (particularly part c) is appropriate for Basic Algebra. 

Algebraic Fractions Simplified 12  With both multiplication and division of fractions with binomials in x, we can simplify by cancelling factors from both numerator and denominator.  
Algebraic Fractions Simplified 13  With both multiplication and division of fractions with binomials in x, we can simplify by cancelling factors from both numerator and denominator.  
Algebraic Fractions Simplified 14  With both multiplication and division of fractions with binomials in x, we can simplify by cancelling factors from both numerator and denominator.  
Algebraic Fractions Simplified 15  With both addition and subtraction of fractions with binomials in x, we can sum (or take a difference) by obtaining a common denominator.  
Algebraic Fractions Simplified 16  With both addition and subtraction of fractions with binomials in x, we can sum (or take a difference) by obtaining a common denominator.  
Algebraic Fractions Simplified 17  With both addition and subtraction of fractions with binomials in x, we obtain a common denominator. You should know that a²  b² = (ab)(a+b). 

Analytic Geoemtry  Circles Problem Set 44  Practice with Advanced Algebra and conic sections, namely, circles. Given an equation, find the equation of a line tangent to the circle at a given point. Or, given a radius and center, determine the equation of the circle. 

Analytic Geometry  Ellipses Problem Set 48  Given equations of ellipses, determine center, foci, vertices, eccentricity, and length of axes (major and minor).  
Analytic Geometry  Ellipses, Problem Set 49  Given parameters of an ellipse, determine the equation. Or, given an equation for an ellipse, determine the parameters (foci, center, eccentricity, etc.) 

Analytic Geometry  Hyperbolas, Problem Set 50  Given parameters of a hyperbola (foci, eccentricity, etc.) determine other parameters or the equation. Or, given an equation for a hyperbola, determine various parameters. 

Area of Square 14013 Version E  Given vertices of a square in Cartesian Coordinates, find the area of the square. This is Version E (explanatory). 

Area of Square 14013 Version F  Given vertices of a square in Cartesian Coordinates, find the area of the square. This is Version F (fast and fleeting). 

Area of Square 14013 Version G  Given vertices of a square in Cartesian Coordinates, find the area of the square. This is Version G (good and general). 

Area of Square 14014 Version E  Given only two vertices of a square in Cartesian Coordinates, find the area of the square. This is Version E (explanatory). 

Area of Square 14014 Version F  Given only two vertices of a square in Cartesian Coordinates, find the area of the square. This is Version F (fast). 

Area of Square 14014 Version G  Given two vertices of a square in Cartesian Coordinates, find the area of the square. This is Version G (general). 

Biquadratic Equations  These fourthorder polynomial problems resemble quadratics. We solve with a basic substitution for x². 

Conic Sections: Circles 12  Four problems (I work two of them); we graph circles from equations.  
Conic Sections: Circles 13  We look for tangent lines to a circle in problem 1,; in problem 3 we calculate center, radius, and equation of a circle.  
Cubic Inequality in One Variable  Consider 2x^3 + 5x^2 is less than or equal to 12x. We solve for the values of x that make the statement true. 

Divide binomials in fractions for Advanced Algebra  You need to understand the division of binomials topandbottom in fractional expressions to venture future into the language of mathematics.  
Dividing "complex" fraction 303  We walk through division of a "complex" fraction with a fraction divided by another fraction, with numerator and denominator each a polynomial in two variables.  
Evaluate (1.04) to 3rd power Version E  We use the binomial theorem (expansion) to calculate (1.04) cubed. This is version E, the elaborate, expansive, explanatory version. 

Evaluate (1.04) to 3rd power Version F  We use the binomial theorem (expansion) to calculate (1.04) cubed. This is version F, the fast, furious, and fun version. 

Evaluate (1.04) to 3rd power Version G  We use the binomial theorem (expansion) to calculate (1.04) cubed. This is version G, the generally good, gradelevel version. 

Exercises of Equations 21  We solve for x (or y or x) using a variety of techniques. In this problem set we have lots of binomials and linear factors. 

Function Domains 05  To stay in the real numbers we cannot take an even root of a negative value, nor can we divide by zero.  
Geometric Progression 14012 Version E  We are to find three (consecutive) numbers in a Geometric Progression whose product is 216 and whose sum is 19. This is version E, the explanatory version. 

Geometric Progression 14012 Version F  We are to find three (consecutive) numbers in a Geometric Progression whose product is 216 and whose sum is 19. This is version F, the fast and fun version. 

Geometric Progression 14012 Version G  We are to find three (consecutive) numbers in a Geometric Progression whose product is 216 and whose sum is 19. This is version G, the general, gradelevel version. 

Graph Inequalities in 2 Variables  Graph linear inequalities in two variables. I like to picture equalities and equations of lines. 

Graph Inequalities in Two Variables  Here we graph solutions to linear inequalities.  
Inequalities in Two Variables  Six little problems in rectangular or Cartesian coordinates. These are linear functions with x and y raised to the first power. 

Inequalities in Two Variables 07  We use equalities to graph these inequalities. These problems use linear relationships. 

Inequality in One Variable with Absolute Value  This simple problem is appropriate toward the end of a course in Basic Algebra.  
Inverses of Functions  Six little problems, you get three to do on your own. We look at inverses of functions of x, as y=f(x) and the inverse is the mirror image about the x=y line. 

Logarithmic Exercises  We solve basic log problems in these exercises.  
Multiplication and Division of Complex Numbers  Multiplication and division of complex numbers uses much of "regular" arithmetic and algebra. Division employs the "complex conjugate" of the denominator. 

Multiply and Divide Complex Numbers  Division of complex numbers involves multiplication that employs the complex conjugate of the divisor.  
Nonlinear Equations  Here we solve nonlinear systems of equations. We have variables raised to powers other than one; we have x² and y² terms. 

Polynomial Division  Quotients and remainders result from these division problems where both dividend and divisor are polynomials in one variable.  
Polynomial Division 09  Division of polynomials can be done with a process that looks just like "long division."  
Problem 261  Domain of Quotient Function  With a quotient of f(x)/g(x) we can easily find the domain of the new function. Note that at one point Mr. X mistakenly calls "the square root of x" function "xsquared." You'll see. 

Problem 262  Graph a Simple Parabola  We walk through a simple sketch of a graph of y = x²  5. The parabola is a conic section. 

Problem 265  Graph Circles from Equations  We look at two circles in Cartesian coordinates from their equations in x and y. We employ desmos.com to see the graphs. 

Problem 266  Graph a Parabola  As y = f(x), we have a simple quadratic that graphs to a parabola. We employ y = a(xh)² + k. 

Problem 267  Parabola as x = f(y)  We look at a "sideways" function with the relation x = f(y) instead of our usual y = f(x).  
Problem 268  Sketch graph of Ellipse  We look at the conic section Ellipse with an equation in x² and y².  
Problem 269  Sketch an Ellipse  For an ellipse centered at (h,k) we have equations in x² and y². We look at a² and b² to determine whether the major axis is horizontal or vertical. 

Problem 270  Graph Ellipse  We complete the square to build the equation of an ellipse prior to graphing it.  
Problem 271  Graph Ellipse  From an equation in x² and y² we complete squares of binomials and build the equation for the ellipse before graphing it.  
Problem 272  Fora conic section (ellipse) with a given center and vertices, and a relation between a and c, sketch the graph and find the equation of the ellipse.  
Quiz, Algebra 14015 Version E  A customer can buy one shirt for x dollars, and additional shirts cost four dollars less than the first. Find the expression that describes the function for the cost of n shirts purchased. This is Version E. 

Quiz, Algebra 14015 Version F  A customer can buy one shirt for x dollars, and additional shirts cost four dollars less than the first. Find the expression that describes the function for the cost of n shirts purchased. This is Version F (shortest). 

Quiz, Algebra 14015 Version G  A customer can buy one shirt for x dollars, and additional shirts cost four dollars less than the first. Find the expression that describes the function for the cost of n shirts purchased. This is Version G (general). 

Simplify Algebraic Fractions  Twelve problems, six of them "worked." We factor out from numerator and denominator common factors, most often linear factors. This is also a Problem Set in Basic Algebra. 

Simplify Algebraic Fractions 17  We simplify algebraic factors by canceling common factors top and bottom. We also divide by multiplying by the reciprocal of the divisor. 

Solve a cubic polynomial for zeroes (roots)  Set y = x^3 + x^2  6x = 0 and solve.  
Solve a cubic polynomial in one variable for zeroes  Solve y = 5x^3  6x^2  8x = 0.  
Solve a fourthorder polynomial for zeroes (roots)  Solve y = x^4  4x^3  5x^2 =0  
Solve Two Eqns. in 3 Unknowns  Solve two equations in three unknowns: x, y and z. We employ augmented matrices. 

Solving Basic Quadratic Equations 27  This is generally considered Basic Algebra, but it begins the foundation for a journey into Advanced Algebra.  
Solving Basic Quadratic Equations 28  Factor quadratics of the form ax²  bx + c = 0.  
Solving Basic Quadratic Equations 29  While this is still Basic Algebra, it underlies our work in Advanced Algebra. We factor secondorder polynomials into linear factors. 

Solving Basic Quadratic Equations 30  Factoring quadratics with coefficients with many factors gives us reason to scratch our heads just a little. Of course, you may always use the Quadratic Formula to solve ax² + bx + c = 0. 

Solving Basic Quadratic Equations 31  A variety of techniques are used to factor quadratics.  
Solving Basic Quadratic Equations 32  More factoring of quadratics. Practice and knowing facts of arithmetic are key. We look briefly at a graph from Wolfram Alpha. 

Solving Basic Quadratic Equations 33  Not every quadratic equation factors into neat binomials or linear factors. In such cases we use the Quadratic Formula. 

Solving Basic Quadratic Equations 34  Factor, if possible. Build experience to see factorizations when they're possible. The Quadratic Formula will always work on a quadratic equation. 

Solving Basic Quadratic Equations 35  We now emphasize the Quadratic Formula, which always serves to solve for the roots of an equation of the form ax² + bx + c = 0.  
Solving Basic Quadratic Equations 36  The Quadratic Formula, which always serves to solve for the roots of an equation of the form ax² + bx + c = 0, should be practiced repeatedly.  
Solving Basic Quadratic Equations 37  A final problem set before embarking on genuine Advanced Algebra with complex answers. This problem set is still entirely "real." 

Solving Basic Quadratic Equations 38  With a negative discriminant we obtain complex roots for the quadratic. Those complex roots always come in pairs. 

Solving Basic Quadratic Equations 39  Twentyfive problems, all solved visually without explanation. You should work these problems before watching the video. 

Solving Basic Quadratic Equations 40  You never know when the discriminant of a quadratic will be negative. Be prepared for complex roots. 

Solving Basic Quadratic Equations 41  Factor when you can, but reliance on the Quadratic Formula is best, particularly with negative discriminants and complex roots. 