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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.

Polynomial Degrees and Definition of a Field [03/02/1998]
The degree of polynomials added together, and definition of a field.

Polynomial Division Compared with Long Division [02/01/2009]
I'm having difficulty grasping the concept of polynomial division.

Polynomial Expansion [02/06/2003]
What's the general formula for things like (a+b+c)^2; (a+b+c+d)^3; (a+ b+c+d+e)^4; (...n+1 terms...)^nth power?

Polynomial Factoring Rules [04/02/1997]
How do I apply the polynomial factoring rules to t^21+1 and 25y^2-144 = 0?

A Polynomial in Three Variables with Few Integer Solutions [03/12/2011]
A student seeks proof that a polynomial in n and two other variables has no integer solutions. After a little insight from modular arithmetic and a lot of searching with a computer algebra system, Doctor Vogler turns up many solutions.

Polynomial Long Division [12/03/2001]
Why in some questions (e.g. b^9+6b^6+b^4+9b^3+4b+8 by b^3+4) do you need to add place holders?

Polynomial Long Division [03/17/2004]
Use long divion to divide (2x - 3) into 4x^4 - x^2 - 2x + 1. I really need help in doing this.

Polynomial Problems [12/4/1995]
1. Let m,n and o be the 3 distinct roots of x^3 + ax + b = 0. 2. Compute (m-n)^2(n-o)^2(o-m)^2 in terms of a and b.... 3. Solve 2x^3 - 3x^2 + 1 = 0....

Polynomial Roots [6/20/1996]
Is there a reliable method to find polynomial roots?

Polynomials of the Fifth Degree and Above [07/28/2001]
I know how to find the root of a polynomial of the form: ax^2+bx+c=0. But what about a polynomial of the third degree?

Proof of a Positive and Infinitely Small Polynomial [4/10/1995]
Prove that there exists a two variable polynomial W(x,y) such that for any x and y it is always positive but at the same time infinitely small.

Proof of Successive Differences and the Degree of a Polynomial [06/26/2007]
Given a sequence of numbers, I know that by finding successive differences between terms I eventually get a constant difference, and that the number of differences needed to get to the constant is the degree of the polynomial that defines the sequence. Can you prove why that works?

Proof That 3 = 0? [10/28/2006]
Starting with x^2 + x + 1 = 0 and using algebraic manipulations, I seem to be showing that 3 = 0. Where am I going wrong?

Proving that f = O(g) Whenever deg(f) <= deg(g) [04/20/2010]
A student seeks formal proof that f(n) = O(g(n)) whenever f(n) and g(n) are polynomials with deg(f) <= deg(g). Doctor Vogler suggests one method for working up to a proof, then obliges with a general strategy that makes use of the sum of the absolute values of all of the coefficients of f(n).

Quadratic Equation [7/10/1996]
Why is a quadratic equation "quadratic"?

Quadratic Equations [12/3/1995]
My students want to know why an equation of the second degree is referred to as a quadratic equation. What does the prefix quad have to do with second degree equations?

Quadratic Formulas, Equations, Parabolas, Graphing [6/16/1996]
Given 6x^2 + x - 1 = 0, how do I find the roots, the vertex, some coordinates, and from these graph it?

Quadratic Function [05/04/2001]
What are the effects of changing the values of a, b, and c in a quadratic function?

Quadratic Inequalities [03/18/2003]
When I solve (x^2 + 1)/4 greater than or equal to (x + 2)/2 , of my final two answers, only one works.

Quadratic Interpolation [03/27/2003]
How do I come up with this polynomial? y = [(x - x2)(x - x3)]/[(x1 - x2)(x1 - x3)]*y1 + [(x - x1)(x - x3)]/[(x2 - x1)(x2 - x3)]*y2 + [(x - x1)(x - x2)]/ [(x3 - x1)(x3 - x2)]*y3.

Quadratics with Odd Coefficients [06/24/2003]
Show that quadratics with odd coefficients have no rational roots.

Quadrinomials [02/26/2001]
What is a quadrinomial, and how is it used?

Quick Way to Expand Binomials Raised to Powers [11/03/2006]
For a binomial expansion like (x+y)^5, I have an easy way to calculate the coefficients without using the standard factorial method or Pascal's triangle.

Quick Way to Expand Large Polynomial Expressions [05/27/2008]
Find the first three terms of the expansion of (x-2)^4*(x+1)^8.

Rational Expressions [01/06/2002]
How do I know when I need to factor and when I don't, and if there are the same polynomials, trinomials, etc., can I cancel them out at all times? What are the restrictions on simplified expressions?

Rationalizing Denominators with Cube Roots [06/26/2008]
How do you rationalize the denominator of a fraction with multiple cube roots like 1/[cubrt(6) + cubrt(9) + cubrt(4)]?

Rational Root Theorem [08/27/1998]
List all possible rational zeros of the each function, then determine the rational zeros: f(x) = x^3 - 4x^2 + x + 2.

Rational Root Theorem [03/02/1999]
Find all possible rational roots of 4x^3 + 3x^2 + 6x + 10.

Rearranging a Polynomial [02/23/2002]
How do you do this question: 2n^2 + y^2 = 66^2 ?

Reducing Algebraic Fractions by Cancelling [03/16/2004]
When reducing fractions, how come you can cancel factors but you can't cancel terms? Can you please explain, hopefully with examples too?

Remainder and Factor Theorem [01/29/1998]
When the polynomial p(x) is divided by (x-1), the remainder is 5 and when p(x) is divided by (x-2), the remainder is 7. Find the remainder when p(x) is divided by (x-1)(x-2).

Restrictions [04/12/2003]
What is the point of restrictions if I can create any one that I want?

Root Multiplicity and Polynomial Functions [11/16/1997]
What effect does multiplicity [e.g. (x+1)(x-2)^2 where -1 has a multiplicity of 1 and 2 of 2] have on a polynomial function?

Roots and the Bisection Method [08/01/1998]
What is the Bisection Method? How would I use it to find the roots of a polynomial?

Roots of a Cubic Equation [05/14/1997]
Find the roots of the cubic equation x^3 = 15x+4 using Cardona's formula.

Series Expansion of 1/(1-x) [08/01/1998]
Can you explain the series expansion identity 1/(1-x) = 1 + x + x^2 + x^3 + ... ? In what region does it converge?

Show that x = 2 [08/29/2001]
Given that: x=(10+(108)^(1/2))^(1/3)+(10-(108)^(1/2))^(1/3), show that x = 2 without using a calculator.

Sign Diagrams [05/20/2002]
I have a diagram in my textbook that I don't understand. What is it supposed to be telling me?

Signs of Roots of 6th-Degree Polynomials [12/12/2000]
When can a 6th-degree polynomial have two positive real roots, two negative real roots, and two imaginary roots? Does it have to do with Descartes' Rule of Signs?

Simplifying Algebraic Fractions [05/01/2008]
To reduce algebraic fractions, be sure to factor first and then cancel, and be careful about canceling terms that are being added and not multiplied, a very common mistake for students starting algebra.

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