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Dr. Math FAQ:
order of operations
Browse High School Polynomials
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Completing the square.
- Polynomial and Remainder [02/19/1998]
An unknown polynomial f(x) of degree 37 yields a remainder of 1 when
divided by x-1...
- 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 =
- A Polynomial Has Third Degree, and Symmetry ... [07/02/2013]
Given the roots r, s, and t of a third-degree polynomial in one variable, a teacher
struggles to find an expression in terms of its coefficients for (1 + r^3)(1 +
s^3)(1 + t^3). Doctor Jacques exploits the function's symmetry and invokes
Viete's formulas to show the way.
- 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 =
- 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
- 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
- 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
- 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
- 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
- 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?