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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.

Functional Similarities: Division by (x -c), and Evaluation at x = c [06/06/2016]
An adult notices similarities when evaluating a function and synthetically dividing it. Doctor Peterson clarifies the relationship between the two methods, introducing the Remainder Theorem along the way — then extends the insight.

Gaussian Elimination [01/11/2000]
How can you solve a system of linear equations using Gaussian elimination?

General Expression for Partial Fractions [02/09/1999]
Is there a general expression for converting a polynomial fraction into partial fraction form?

Graphing an Equation [11/08/1997]
Please explain how to graph this equation: y = a + b(x) + c(x^2) + ...

Graphing Multivariable Polynomials [02/20/2005]
I have a few questions about multivariable polynomials. Say we have the polynomial x + y + z, can this be graphed? I also really don't know anything about the z axis or how to graph on it.

Graphing Polynomial Functions in Factored Form [07/07/2004]
When graphing polynomial functions, why do factors of even multiplicity cause the graph to not cross the x-axis, while factors of odd multiplicity cause it to cross?

Graphing Quadratic Polynomials [02/07/2001]
Can you give me a simple explanation of how to graph quadratic polynomials like y = x^2 - 8x + 15?

How Are Functions and Expressions Related? [07/09/2004]
What is the relationship between a function and an expression? I don't see any relationship, they are two completely different things.

How does e to the pi i = -1? [1/6/1995]
I want to know how "e to the pi i" = -1. Can you show me how that may be evaluated and the type of formulas that are involved?

How Do You Know When to Stop Factorizing? [03/09/2010]
How do you know when you've completely finished factoring a polynomial? Doctor Vogler invokes Bezout's Theorem, and walks through a couple examples.

How to Factor Polynomials [3/14/1995]
Hi, I am having trouble factoring the following polynomials: 6a^2y^2+5ay^2-6y^2 and 3ax^2+7ax+2a

Importance of Constant Term in Descartes' Rule of Signs? [05/03/2006]
One of my students seems to have come up with an exception to Descartes' Rule of Signs with the function p(x) = x^3 + x. Am I missing something?

Inconsistently Improper? [05/07/2017]
A teen tries to reconcile why we call an algebraic fraction improper if it evaluates to a proper fraction. Doctor Peterson clarifies how the scope of the terms "proper" and "improper" differs when applied to arithmetic versus algebra.

Integral of a Polynomial [7/15/1996]
What is the integral of (5x-1) to the power of 6?

Introduction to Algebraic Fractions [05/29/2008]
What are algebraic fractions and how do they work?

Irreducible Polynomials [06/06/2001]
If provided with an irreducible polynomial, how can you prove that it is indeed irreducible? E.g. the polynomial x^8+x^4+x^3+x+1 (Hex: x'11B').

Isolating a Variable in a Tricky Place, or Raised to a Power [10/01/2010]
A student struggles to isolate terms in a kinematics equation. To express the formula in terms of acceleration, Doctor Ian suggests substituting; then points out its quadratic nature to solve it with respect to time.

Is (x^(1/2))^4 a polynomial? [06/10/2002]
Is (x^(1/2))^4 a polynomial or not?

Lagrange Interpolation [08/20/2003]
I am trying to learn Lagrange interpolation.

Largest Positive Integer [11/04/2002]
What is the largest positive integer n for which n^3 + 100 is divisible by n+10?

Logarithm Equation [01/22/2002]
Is there a way to solve this equation: 2x^2-1 = ln(2x^2) ?

Logarithms of the Zeros of a Quadratic [03/24/2002]
If p and q are the zeros of 2x^2-mx+1=0, what is the value of log(base 2)p + log(base 2)q?

Long Division of Polynomials [6/26/1995]
How do you use long division to solve polynomials with remainders?

Long Division with Polynomials [12/11/2007]
How do you divide (2x^6 + 5x^4 - x^3 + 1) by (x^2 + x + 1) either by synthetic or long division?

Meaning of Term in Algebra [09/16/2007]
Why is it that when you multiply or divide two numbers or variables they become one term (such as 2*x becoming 2x), but when you add or subtract two numbers or variables they are still two terms (such as 2 + x)?

Multinomial Coefficients [02/02/2004]
How can I find the coefficient of a particular term in the expansion of (x+y+z)^9, without actually expanding it?

Multinomial Theorem and Coefficients of Polynomial Expansions [06/24/2007]
Could you explain how to use the multinomial theorem to find the coefficients of various terms in the expansion of (x+y+z+a+b)^4?

Multiplication of Polynomials [3/20/1996]
I need to find the product for these two problems: 7(4m-3)(2m+1) and (2x- 1)(3x exp.5 -2x exp.3+x exp.2-2x+3)

Multiplying Binomials and Other Polynomials [04/16/2004]
I have a hard time multiplying binomials properly, such as (2x + 5)(x + 7). I do not know what order the problem should be solved in.

Multiplying Polynomials [01/29/2001]
Why isn't (3x+2) squared equal to 9x squared + 4?

Names of Polynomials [11/25/1997]
Are polynomials with higher degree than 5 named? What are they called?

Names of Polynomials with More Than Three Terms [03/25/2004]
What are the official, mathematical names for polynomials with more than three terms? For example, a polynomial with one term is a monomial, with two terms is a binomial, and with three terms is a trinomial. What is a polynomial with more than three terms called?

Negative Coefficients, Subtraction -- and Combining Like Terms? [09/21/2009]
How do I combine like terms with an expression like -9x - y + 1 + 5y + 5x - 10?

Negative Numbers Combined with Exponentials [03/09/2001]
Why in the order of operations is negation a multiplication done after exponentiation, rather than as a part of the base? What about polynomials?

Negative Remainders in Arithmetic and Algebra: a Difference of Degrees [05/14/2010]
An algebra teacher wonders how to explain the existence of negative polynomial remainders to students accustomed to positive remainders. Doctor Peterson clarifies the difference between the remainders of arithmetic and those of algebra, underscoring polynomial degree.

Nonlinear Factors [10/12/1998]
I have been told that factoring the sum of two squared numbers is not possible, but factoring x^4 + 4 is possible. Please explain.

Non-Linear Simultaneous Equations [12/07/2003]
I have four simultaneous equations with four variables, but each one has a non-linear term in it (e.g., ab + c + d = 3). So none of the usual techniques will work! How can I solve the system?

Number of Roots of Polynomial with Radical [02/05/2002]
Solve this equation:r-9(r^(1/2))+8 = 0.

The Number of Terms in an Expression [07/12/2006]
Why is 4x + 4y + 4z three terms but 4(x + y + z) only one term?

Number of Terms in a Polynomial Expansion [12/01/2005]
How would you calculate the total number of terms in the simplified expansion of (a+b+c+d)^10? Is there a general formula for this?

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