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High School Archive

Browse High School Polynomials
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Completing the square.
    Quadratic equations.

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Simplifying Variable Expressions [09/07/2006]

How do I simplify an expression like -16[42 - 3(c + 2)] - 16? How does order of operations work in this sort of problem?

Slicing Up a Circle [03/22/2001]

Find a formula that will give the maximum number of pieces with n number of straight slices of the circle.

Solving (2x+3)^4 [01/22/2001]

Pascal's triangle, and a recursive formula.

Solving a Multi-Variable Formula or Equation for One Variable [07/24/2008]

The keys to solving an equation like y^2 + 3yz - 8z - 4x = 0 for a single variable, such as z, are to group terms, take out a common factor, and avoid common simplifying mistakes.

Solving a Polynomial of Any Degree [06/02/2003]

Combining Newton's method with the bisection method.

Solving a Quartic Equation with Substitutions [11/02/2004]

I'm trying to solve y(y + 1)(y + 2)(y + 3) = 7920. I've multiplied it out to get y^4 + 6y^3 + 11y^2 + 6y - 7920 = 0, but I can't find a way to factor or solve that equation. Can you help?

Solving a Quartic Polynomial using a Trig Substitution [12/27/2001]

I need help solving this system: y/sqrt(y^2+64) = (x+y)/25; x/sqrt(x^ 2+64) = (x+y)/20.

Solving Cubics (3rd Degree Polynomials) [12/15/1996]

How do you solve problems of the type ax^3+bx^2+cx+d = 0 ?

Solving Linear Equations without PEMDAS [08/13/2002]

Is there a linear equation that can be solved using both the PEMDAS order of operations and NOT using the order of operations and still come out with the same answer?

Solving Polynomials [6/16/1996]

(x^2 + 2x)/3 + 3/(x^2 + 2x) = 26/5

Solving Third-Degree Equations [09/24/2001]

I'm trying to solve x^3 + 2x^2 - 3x - 4 = 0. I need to take the square root of 4p^3 + 27q^2, which is -272. I'm looking for the real solutions, and there are three of them.

Square Roots and Limits [03/12/2003]

Given a value of x, find the value of the expression sqrt(x sqrt(x sqrt(x...

The Sum of a Number and Its Reciprocal [11/04/2003]

I need a formal proof showing that the sum of a positive number and its reciprocal is at least 2. I can prove it algebraically, but I need a visual justification.

Sum of Powers of Roots of Polynomial Equations [01/02/2007]

Is there a quick trick to find the sum of powers of roots for third degree polynomial equations? For example, given roots a, b, and c, find a^3 + b^3 + c^3 or a^4 + b^4 + c^4.

Sum of Roots of n-degree Polynomial [01/24/2001]

Prove that for all n roots of unity (z^n = 1), the sum of n roots is zero for n greater than 1.

Symmetric Polynomials [07/31/1997]

How can it be proved that any symmetric polynomial can be expressed as an elementary symmetric polynomial?

Synthetic Division [08/05/1997]

Please explain synthetic division.

Synthetic Division [12/22/1996]

How does synthetic division work?

Synthetic Division in Factoring [04/18/2002]

Factor completely: 2x^3 + 17x^2 + 58x + 25.

Synthetic Division when Coefficent of Linear Term is Not 1 [12/15/2005]

How can I do synthetic division if I am dividing by something like 4x + 5? I only know how to do it for divisors that start with x, like x - 3.

Theorem About Sum and Product of Quadratic Roots [03/07/1998]

Explain why 1/2 and 3/4 are NOT the roots of 0 = 4x^2 + 5x + 8.

Three Polynomial Questions [8/7/1995]

Three interesting polynomial questions from a past trial HSC paper...

Trinomial Expansion [02/23/2003]

How can I find the number of terms of (a + b + c)^140?

Trinomials Squares of Binomials [04/10/2003]

Is the trinomial x squared -2x + 1 the square of a binomial?

Two Methods of Factoring Quadratics [04/14/1998]

Could you describe how to factor quadratic trinomials?

Undefined and Indeterminable ... at the Same Time? [09/05/2010]

A student wonders whether the labels "undefined" and "indeterminate form" could apply to one and the same expression. Doctor Vogler considers several expressions, functions, and limits to distinguish the different contexts that call for such terminology.

'Un Distributing' Polynomials [01/11/2003]

How can you factor: y^4 + 14y^2 + 49 - x^2 ?

Un-nesting Radicals [03/16/2001]

How can I simplify sqrt(2+sqrt[3])?

Using Cardan's Formula to find Real Roots [04/13/2000]

How can I show that, although the cubic equation x^3 - 6x = 4 has three real solutions, Cardan's formula can find them by subtracting appropriate cube roots of complex numbers?

Visual Representation of (a+b)(a-b) [03/22/2003]

Is there a way to model (a+b) * (a-b)?

What's His Street Address? How Many Neighbors Does He Have? [09/29/2010]

A student struggles with a word problem that asks for specific sums of counting numbers. Three different doctors weigh in with increasingly sophisticated and comprehensive problem-solving approaches: programming spreadsheet formulas; applying combinatorics; and invoking quadratic Diophantine and Pell equations.

Why Algebraic Expressions with Parentheses? [03/18/2003]

How would you write these algebraic expressions without parentheses? -(2x-3y-6) and -(5x-13y-1).

Why Do We Use Order of Operations? [09/18/2008]

Why is it necessary to use order of operations? Why can't you just write a calculation from left to right, so 5+2^3*2 is just 2^3*2+5? Doctor Peterson shows how order of operations makes writing and understanding algebraic expressions quite simple.

Why Factor? [08/18/2001]

I've been explained the how's, the where's, the what's, but not the WHY. What is the practical use of factoring?

Why Factor Polynomials? [02/20/1998]

Why is factoring polynomials important for our everyday lives?

Why FOIL? [11/21/2002]

How can I explain to the class the importance of learning FOIL?

Zeros of Polynomials [9/24/1996]

Given that 1 is a zero of the equation P(x) = x^3-5x^2+17x-13, find all other zeros.

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