Algebra Problems

Problem 1

Find the line through (1,-6) parallel to y=3x-1.  Find the line through (-3,1) which is perpendicular to 2x+3y=8.

Problem 2

What point on 5x-12y=8 is closest to (1,3)?

Problem 3

Graph the equation (x^2)+(y^2)+10x-8y=23.

Problem 4

Expand (((2x-3))^5)

Problem 5

Find all positive integer solutions to (({{n+3}, {3}}))+(({{n+2}, {3}}))+(({{n+1}, {3}}))=111.

Problem 6

Graph the set of points (b,c) for which the equation (x^2)+b x+c=0 has non-real roots.

Problem 7

Find the exact value of (((((Sqrt[3])/2)-(1/2)i))^10)

Problem 8

If z represents a complex number, how many solutions are there to |z((|)^2)-15=(4+2i)(Overscript[z, -]) ?

Problem 9

How many real solutions are there to (x^5)+2(x^3)+3x+10=0?

Problem 10

How many real solutions are there to (x^5)-10x+1=0?

Problem 11

Find a polynomial with integer coefficients for which x=(Sqrt[11])-(Sqrt[3]) is a solution.

Problem 12

Suppose r is a solution to the equation (x^3)-x+1=0.  Find a polynomial equation with integer coefficients for which 2r-1 is a solution.

Problem 13

Find the quotient when (x^2)-3x+7 is divided into (x^5)-3(x^2)+x+1.

Problem 14

Find the remainder when (x^2)+2 is divided into (x^20)+(x^7)+2(x^3)-3x+1

Problem 15

Suppose that (Subscript[r, 1]), (Subscript[r, 2]), (Subscript[r, 3]), and (Subscript[r, 4]) are the solutions to 2(x^4)-3(x^3)+17(x^2)-37x+8=0 over the complex numbers.  Find ((((Subscript[r, 1])+(Subscript[r, 2])+(Subscript[r, 3])+(Subscript[r, 4])))^3)-10 (Subscript[r, 1])(Subscript[r, 2])(Subscript[r, 3])(Subscript[r, 4]).

Common Algebra Concepts

Equations of Lines (Slope-intercept, Point-Slope)

Parallel and Perpendicular Lines

Equations of Circles

Quadratic Equation, Quadratic Discriminant, Completing the Square

Polynomial Division (quotient and remainder, working “mod” a polynomial)

Binomial Theorem and Pascal’s Triangle

Properties of complex numbers (addition, subtraction, multiplication, division, absolute value, conjugate)

Polar form for a complex number/Geometric multiplication

Number of solutions to a polynomial equation (Fundamental Theorem of Algebra, Conjugate Theorem, Rational Roots Theorem, Intermediate Value Theorem, Applications of Rolle’s Theorem, Descartes Rule of Signs)

Properties of roots of polynomials, finding an equation a given number satisfies