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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.

A Cubic Curve, Its Double Root, and Its Integer Solutions [02/27/2011]
A student seeks integer solutions to (x^2 + y)(x + y^2) = (x - y)^3. To oblige, Doctor Vogler parameterizes the equation of this twisted curve.

Cubic Equations [01/19/1998]
How would you solve a problem like 3x^3+x^2+15x+27 = 0?

Cubic Equations [05/17/2001]
How do you solve cubic equations? Is there a general formula?

Cubic Equations in One Formula [11/27/2001]
I know the formula for solving quadratic equations of the type ax^2+bx+c=0, but is there a formula for cubic equations?

Decomposing Fractions [04/08/2009]
Finding the sum of 1/(1*2) + 1/(2*3) + 1/(3*4) + ... + 1/(29*30) by decomposing into partial fractions.

Defining Quadratic Formula [01/21/2002]
Taking successive differences in a sequence.

Definition of a Polynomial [12/04/1997]
Is just x a polynomial?

Definition of Monomial [04/26/2004]
Do monomials include quotients of variables? For example, would x/y or xy^-2 be considered a monomial?

Definitions of Monomials and Polynomials [03/11/2004]
Why is 1/x considered not to be a monomial? It could be written as x^(-1) which is just one term. My math text says that it is NOT a monomial but it does not say why.

Degree of a Constant [12/02/2002]
Why is the degree of a constant zero?

Degree of a Polynomial [12/31/1994]
If the degree of a monomial is the sum of the exponents of its variables, so that x^3*y^2 is degree 5, what happens in the case of x^3*y^(-2)? Is it degree 1 or would you apply absolute value and also call it degree 5?

Degree of a Rational Function [06/19/2003]
Given: f(x) = p(x)/q(x). Prove that if the highest degree of p(x) is smaller than the highest degree of q(x), the horizontal is y=0.

Degree of Constant Function [11/08/2001]
We think F(x) = 1x^0 is not a polynomial function (because polynomials shouldn't have discontinuities), but F(x) = 1 is a polynomial. And F(x) = 1 still has degree 0 but for reasons we can't explain.

Degree of Zero [02/25/2003]
What is the degree of zero?

Descartes - Rule of Signs [06/26/1998]
I'm trying to find the number of positive and negative real zeros for a polynomial...

Descartes' Rule of Signs and Complex Roots [08/04/1998]
Prove that if p and q are real and q is not equal to 0, the equation x^3 + px +q = 0 has two imaginary roots.

Difference between Two Cubes [11/09/2001]
Factoring a^3 - b^3.

Difference of Squares of Two Integers [7/2/1996]
Which positive integers can be written as the difference of the squares of two integers?

Difference of Two Cubes [11/18/1996]
What two positive integers satisfy the condition that the difference of their cubes is 56,765?

Difference of Two Squares [09/04/2002]
I understand how to get the difference of two squares but how would you get the equation when you have the answer?

Difference Tables [05/09/1997]
Why is it that with almost any values for the difference table, a common number is always reached to determine the degree of the equation?

A Different Binomial Expansion Theorem [07/23/1998]
I'm looking for an alternate version of the famous binomial expansion theorem.

The Discriminant of Quadratic Equations [08/06/1998]
Can you explain more about the discriminant, b^2 - 4ac? Why is it sometimes represented by a triangle (delta)?

Dividing Polynomials [03/06/1997]
The polynomial p(x) with integer coefficients satisfies (1) if p(x) is divided by x^2-4x+3, the remainder is 65x-68, and (2) if p(x) is divided by x^2+6x-7, the remainder is -5x+a. Find a.

Dividing Polynomials [8/18/1996]
How do I simplify ([a^3-6a^2+8a]/5)/([2a-4]/[10a-40])?

Dividing Polynomials of Different Degrees and Coefficients [11/22/2015]
How do you divide polynomials when their degrees and coefficients do not match? Comparing polynomial division to standard long division, Doctor Ian provides some guidance.

Dividing two Polynomials [6/28/1996]
Divide these two Polynomials: 12(x^2 - 4x + 3) / x^3 - 3x^2 -x >+ 3.

Division of Unknown Polynomials [3/18/1996]
When a polynomial P(x) is divided by x-1, the remainder is 3. When P(x) is divided by x-2, the remainder is 5. Find the remainder when P(x) is divided by x^2-3x+2.

Eisenstein's Irreducibility Criterion [02/26/2003]
Is Eisenstein's irreducibility criterion necessary or sufficient for a polynomial to be irreducible?

Equations and Factoring [2/5/1996]
Solve. Identify all double roots: 2(r^2 + 1)=5r

Euler's Faith and Folly [03/25/2011]
What did Euler wrongly assume when he first derived pi^2/6 from the infinite sum of 1/n^2? Doctor Jordan reveals two missteps initially committed by the famous mathematician on this now-classic result.

Expanding Binomials and Pascal's Triangle [06/07/1998]
How does expanding a binomial relate to Pascal's Triangle?

Expressions vs. Equations, Explained [04/10/2011]
A student knows how to evaluate expressions and solve equations, but doesn't see the difference between the two. Doctor Ian explains the distinction by way of an analogy to phrases and sentences, illustrating throughout with examples, before leaving the student with a mathematical statement for her to ponder further whether "an equation is an expression with an equals sign."

Expression vs. Equation [03/20/2003]
What is the difference between an expression and an equation?

Factor and Multiply to Check [5/21/1996]
Is x to the second power plus 6x plus 5 factorable?

Factor and Simplify [4/10/1996]
How would you simplify the following problem? (1/x^2- 1/y^2)/(1/x^2+2/xy+1/y^2)

Factor for X; Consecutive Odd Integers [01/19/1998]
(x-2)(x+3) = 6 : solve by factoring for x. Label your variables, write an equation, solve, label answers...

Factoring [12/03/2001]
This problem is giving me a hard time: 6x^2 + 31x + 5.

Factoring 5x^2 + 2x -1 [01/22/2001]
I can't figure out how to factor this: 5x^2 + 2x -1. The best I can do is: (5x+1)^2 + 2(x-1).

Factoring 8 - t^3 [01/07/1998]
Factor the given expression completely: 8 - t^3.

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