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Dr. Math FAQ:
order of operations
Browse High School Polynomials
Stars indicate particularly interesting answers or
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Selected answers to common questions:
Completing the square.
- 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
- 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
- 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
- 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
- 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?
- Order of Operations and Negation in Excel [01/13/2000]
A discussion of why Excel evaluates an expression like -3^2 as (-3)^2
and a look at the mathematical conventions behind the confusion.
- Parabolas Through Pairs of Points [07/13/2002]
Can infinitely many parabolas pass through any two points in a plane?
- Partial Fractions [01/29/1998]
How do I express 3/1-(x^3) in partial fractions?
- Partial Fractions [11/23/2003]
I need to put this fraction into power series formation: (3x^2 - x)/(x^3 - x^2 - x + 1). I've tried to use partial fractions but keep getting stuck.
- Pascal's Triangle and Binomial Expansions [09/01/1997]
I need to use Pascal's triangle to write out the binomial expansion of
something like (X+Y)^6.
- Polynomial and Remainder [02/19/1998]
An unknown polynomial f(x) of degree 37 yields a remainder of 1 when
divided by x-1...