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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.
- 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?
- 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.
- Simplifying Radicals [06/18/2001]
In the following: sin(15) = sqrt(2-sqrt(3))/2 = (sqrt(6)-sqrt(2))/4, how
do you manipulate sqrt(2-sqrt(3))/2 to obtain (sqrt(6)-sqrt(2))/4?
- Simplifying Rational Expressions [06/13/2002]
Simplify (-6x^2 - 4xy + 8x)/(2x).
- Simplifying Square Root within Square Root without Calculator [07/05/2004]
I need to find sqrt[5 + sqrt(2)] without using a calculator. Is there
a general formula for solving a problem like this?
- Simplifying the Square Root of (a + b*sqrt(c)) [07/29/2004]
How do I find more advanced square roots, such as sqrt(11 - 2sqrt(18))?
- Simplifying Variable Expressions [09/07/2006]
How do I simplify an expression like -16[42 - 3(c + 2)] - 16? How does
order of operations work in this sort of problem?
- Slicing Up a Circle [03/22/2001]
Find a formula that will give the maximum number of pieces with n number
of straight slices of the circle.
- Solving (2x+3)^4 [01/22/2001]
Pascal's triangle, and a recursive formula.
- Solving a Multi-Variable Formula or Equation for One Variable [07/24/2008]
The keys to solving an equation like y^2 + 3yz - 8z - 4x = 0 for a
single variable, such as z, are to group terms, take out a common
factor, and avoid common simplifying mistakes.
- Solving a Polynomial of Any Degree [06/02/2003]
Combining Newton's method with the bisection method.
- Solving a Quartic Equation with Substitutions [11/02/2004]
I'm trying to solve y(y + 1)(y + 2)(y + 3) = 7920. I've multiplied it
out to get y^4 + 6y^3 + 11y^2 + 6y - 7920 = 0, but I can't find a way
to factor or solve that equation. Can you help?
- Solving a Quartic Polynomial using a Trig Substitution [12/27/2001]
I need help solving this system: y/sqrt(y^2+64) = (x+y)/25; x/sqrt(x^
2+64) = (x+y)/20.
- Solving Cubics (3rd Degree Polynomials) [12/15/1996]
How do you solve problems of the type ax^3+bx^2+cx+d = 0 ?
- Solving Linear Equations without PEMDAS [08/13/2002]
Is there a linear equation that can be solved using both the PEMDAS
order of operations and NOT using the order of operations and still
come out with the same answer?
- Solving Polynomials [6/16/1996]
(x^2 + 2x)/3 + 3/(x^2 + 2x) = 26/5
- Solving Third-Degree Equations [09/24/2001]
I'm trying to solve x^3 + 2x^2 - 3x - 4 = 0. I need to take the square
root of 4p^3 + 27q^2, which is -272. I'm looking for the real solutions,
and there are three of them.
- Square Roots and Limits [03/12/2003]
Given a value of x, find the value of the expression sqrt(x sqrt(x
- The Sum of a Number and Its Reciprocal [11/04/2003]
I need a formal proof showing that the sum of a positive number and its
reciprocal is at least 2. I can prove it algebraically, but I need a
- Sum of Powers of Roots of Polynomial Equations [01/02/2007]
Is there a quick trick to find the sum of powers of roots for third
degree polynomial equations? For example, given roots a, b, and c,
find a^3 + b^3 + c^3 or a^4 + b^4 + c^4.
- Sum of Roots of n-degree Polynomial [01/24/2001]
Prove that for all n roots of unity (z^n = 1), the sum of n roots is zero
for n greater than 1.
- Symmetric Polynomials [07/31/1997]
How can it be proved that any symmetric polynomial can be expressed as an
elementary symmetric polynomial?
- Synthetic Division [08/05/1997]
Please explain synthetic division.
- Synthetic Division [12/22/1996]
How does synthetic division work?
- Synthetic Division in Factoring [04/18/2002]
Factor completely: 2x^3 + 17x^2 + 58x + 25.
- Synthetic Division when Coefficent of Linear Term is Not 1 [12/15/2005]
How can I do synthetic division if I am dividing by something like 4x
+ 5? I only know how to do it for divisors that start with x, like x - 3.
- Theorem About Sum and Product of Quadratic Roots [03/07/1998]
Explain why 1/2 and 3/4 are NOT the roots of 0 = 4x^2 + 5x + 8.
- Three Polynomial Questions [8/7/1995]
Three interesting polynomial questions from a past trial HSC paper...
- Trinomial Expansion [02/23/2003]
How can I find the number of terms of (a + b + c)^140?
- Trinomials Squares of Binomials [04/10/2003]
Is the trinomial x squared -2x + 1 the square of a binomial?
- Two Methods of Factoring Quadratics [04/14/1998]
Could you describe how to factor quadratic trinomials?
- Undefined and Indeterminable ... at the Same Time? [09/05/2010]
A student wonders whether the labels "undefined" and "indeterminate form" could
apply to one and the same expression. Doctor Vogler considers several expressions,
functions, and limits to distinguish the different contexts that call for such terminology.
- 'Un Distributing' Polynomials [01/11/2003]
How can you factor: y^4 + 14y^2 + 49 - x^2 ?
- Un-nesting Radicals [03/16/2001]
How can I simplify sqrt(2+sqrt)?