Polynomials

• Adding and Subtracting Polynomials [James, 12/08/1996]
  How do you add and subtract polynomials?

• Binomial Expansions and Pascal's Triangle [Fama, 7/10/1996]
  What are binomial expansions and where can they be used?

• Complex Roots [Scott, 11/1/1994]
  We know it is possible to look at the graph of a polynomial and tell a great deal about its real roots by looking at the x-intercepts. What can be discovered about a polynomial's complex roots by looking at the graph? There seem to be some interesting "wiggles" at locations that appear to be related to the "average" of the complex pairs. It appears that the "wiggle" of these graphs is always influenced by the complex roots. What we are trying do is develop a graphing technique that will let us find the complex roots from the real graph. (Contributions by Profs. Conway and Maurer.)

• Do Irrational Roots Always Occur in Conjugate Pairs? [Libby, 03/13/2002]
  A colleague and I have been puzzled by the equation x^3+4x^2+3x+1=0. Upon graphing, there appear to be two imaginary solutions and one real solution...

• Explaining Algebra Concepts and FOIL [Rachel, 08/04/1998]
  Some ideas for learning algebra that may help prevent frustration.

• Factoring Polynomials [Cairns, 11/14/1996]
  Is there a logic behind the factorization of polynomials?

• Factoring Polynomials [Kate, 12/18/1997]
  I have no idea how to factor these problems: 12*13-60+12^2; 11^2-6*11+ 5*11.

• Finding sum formula using sequences of differences [Kyungsoo, 06/28/1998]
  Finding a formula for the sum of the first n fourth powers using sequences of differences.

• How to Factor [Louie, 7/25/1996]
  I have forgotten how to factor - could you give me an example and an explanation?

• I'd like help with long division of polynomials [Mody, 3/4/1995]
  (x^4+4y^4)/(x^2-2xy+2y^2). Please show me carefully, step by step!

• Parent Pulling Her Hair Out [Carol, 03/03/2001]
  Given (x+3)^3 + 2(x+3)^2 - 8(x+3) = 0, I get as far as x^3 + 11x^2 + 21x + 21 = 0 and get stuck.

• Pascal's Triangle and Binomial Coefficients [Michelle, 09/11/2001]
  Why are Pascal's triangle and the binomial coefficients the same?

• Regrouping Polynomials [Jessica, 11/19/2001]
  I'm stuck on problems like a^2x - bx - a^2y + by + a^2z - bz.

• Simplifying Algebraic Expressions [Heidi, 03/23/2002]
  Please explain how to simplify algebraic expressions.

• Degree of a Polynomial [Mitchell, 12/31/1994]
  The degree of the polynomial is the exponent sum of the variables.. x**2 y**3 is a degree 5 but what about x**3 y**-2 Is the exponent sum 1 or is it the absolute value sum and degree 5? I assume it will be the absolute value sum. . .

• How does e to the pi i = -1? [So, 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?

• Formula for Finding Roots [Thompson, 1/20/1995]
  I need to know the formula for finding the roots of any quadratic, cubic, or quartic polynomial.

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

• Proof of a Positive and Infinitely Small Polynomial [Kacprzak, 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.

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

• Three Polynomial Questions [Sharpe, 8/7/1995]
  Three interesting polynomial questions from a past trial HSC paper...

• Polynomial Problems [Howard, 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 = 0....

• Division of Unknown Polynomials [Jacob, 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.

• Multiplication of Polynomials [Sam, 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)

• Factoring a Degree 2 Polynomial [Brandon, 3/21/1996]
  Questions like x^2 - 12x - 85 I don't get these factor things!

• Solving Polynomials [Lisa, 6/16/1996]
  (x^2 + 2x)/3 + 3/(x^2 + 2x) = 26/5

• Polynomial Roots [Handel, 6/20/1996]
  Is there a reliable method to find polynomial roots?

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

• Binomial Theorem by Induction [Turner, 7/14/1996]
  I'm trying to prove the Binomial Theorem by Induction, but I'm having trouble going from the hypothesis step to the n+1 step.

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

• Binomial Theorem [Mathot49, 8/5/1996]
  Find the fourth term of (2a - 6b)^11.

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

• Find Values of x... [Atienza, 9/4/1996]
  Determine all values of x for which (x^2-5x+5)^(x^2-2x-48)=1.

• Factoring Quartics [Atienza, 9/4/1996]
  Can you help me factorize f(x) = x^4 - 6x^3 + 11x^2 - 6x + 1 and solve f(x) = 0?

• Zeros of Polynomials [Barrow, 9/24/1996]
  Given that 1 is a zero of the equation P(x) = x^3-5x^2+17x-13, find all other zeros.

• Finding the Roots of a Function [Thompson, 11/4/1996]
  How do you find the positive roots of x^9+3x^8-5x^3+4x+6=0? Show that x^n- 1=0 has exactly 2 roots if n is even, and only 1 real root if n is odd

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

• Completing the Cube [Smolka, 11/28/1996]
  Is there some method analogous to completing the square for higher dimension polynomials?

• Solving Cubics (3rd Degree Polynomials) [Walden, 12/15/1996]
  How do you solve problems of the type ax^3+bx^2+cx+d = 0 ?

• Synthetic Division [Mcclain, 12/22/1996]
  How does synthetic division work?

• Completing the Square [Holley, 01/14/1997]
  How do you complete the square? Can you use this to factor polynomials?

• Factoring Expressions [Heather, 01/19/1997]
  Factor 3x-21, 5x^2y-15xy^2, 18x^2-27x, and 2a-8b-10.

• Dividing Polynomials [Binnig, 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.

• Polynomial Factoring Rules [Price, 04/02/1997]
  How do I apply the polynomial factoring rules to t^21+1 and 25y^2-144 = 0?

• Algebraic expressions and fractions [Dean, 04/10/1997]
  If X/A+Y/B+Z/C = 1 and A/X+B/Y+C/Z = 0, what is (X^2)/(A^2)+(Y^2)/(B^2)+(Z^2)/(C^2)?

• Finding Polynomials [Elizabeth, 04/29/1997]
  Given a set of points, find the polynomial that fits them.

• Difference Tables [DeJong, 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?

• Roots of a Cubic Equation [Givins, 05/14/1997]
  Find the roots of the cubic equation x^3 = 15x+4 using Cardona's formula.

• Symmetric Polynomials [Stewart, 07/31/1997]
  How can it be proved that any symmetric polynomial can be expressed as an elementary symmetric polynomial?

• Pascal's Triangle and Binomial Expansions [Meghan, 09/01/1997]
  I need to use Pascal's triangle to write out the binomial expansion of something like (X+Y)^6.

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

• Root Multiplicity and Polynomial Functions [Rubinshteyn, 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?

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

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

• Factoring Trinomials - Difference of Two Squares [Church, 12/22/1997]
  How do I factor a perfectly square trinomial and a quadratic trinomial?

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

• Partial Fractions [Yanzhen, 01/29/1998]
  How do I express 3/1-(x^3) in partial fractions?

• Remainder and Factor Theorem [Zhen, 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).

• Adding Polynomials [Sara, 02/15/1998]
  I don't know how to add polynomials. Can you please help me?

• Polynomial and Remainder [Freeman, 02/19/1998]
  An unknown polynomial f(x) of degree 37 yields a remainder of 1 when divided by x-1...

• Why Factor Polynomials? [Wrigley, 02/20/1998]
  Why is factoring polynomials important for our everyday lives?

• Polynomial Degrees and Definition of a Field [Metelli, 03/02/1998]
  The degree of polynomials added together, and definition of a field.

• Factoring Quadratics [Stanfield, 03/05/1998]
  Factoring quadratics that occur in fractions.

• Theorem About Sum and Product of Quadratic Roots [laura, 03/07/1998]
  Explain why 1/2 and 3/4 are NOT the roots of 0 = 4x^2 + 5x + 8.

• Polynomial Basics and Terms [Erin, 05/20/1998]
  I'm having a lot of trouble understanding polynomials.

• Counting Odd Coefficients [Ehsan, 05/27/1998]
  If (1+x)^100 is multiplied out, how many of the coefficients are odd? How would you generalize?

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

• Finding Coefficients of Simultaneous Equations [Emma, 06/24/1998]
  The equations ax^4 + bx^3 + c = 0 and cx^4 + bx^3 + a = 0 have a common root. Find all possible values of b, if a + c = 100.

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

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

• Series Expansion of 1/(1-x) [Edic, 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?

• Roots and the Bisection Method [JCG, 08/01/1998]
  What is the Bisection Method? How would I use it to find the roots of a polynomial?

• Descartes' Rule of Signs and Complex Roots [McDowell, 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.

• Rational Root Theorem [Drummond, 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.

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

• Rational Root Theorem [Nicole, 03/02/1999]
  Find all possible rational roots of 4x^3 + 3x^2 + 6x + 10.

• Binomial Expansions [Jeff, 03/14/1999]
  Find the constant term of the expansion of (x + x^-1)^6.

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

• Signs of Roots of 6th-Degree Polynomials [Christopher, 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?

• Complex Conjugate Roots of Real Polynomials [Swank, 01/11/2001]
  How can I prove that if a polynomial p(x) with real coefficients has a complex number as a root, then its complex conjugate must also be a root?

• Solving (2x+3)^4 [Sadreddini, 01/22/2001]
  Pascal's triangle, and a recursive formula.

• Sum of Roots of n-degree Polynomial [Schulte, 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.

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

• Quadrinomials [Morrill, 02/26/2001]
  What is a quadrinomial, and how is it used?

• Negative Numbers Combined with Exponentials [Ryno, 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?

• Un-nesting Radicals [Tran, 03/16/2001]
  How can I simplify sqrt(2+sqrt[3])?

• Slicing Up a Circle [Don, 03/22/2001]
  Find a formula that will give the maximum number of pieces with n number of straight slices of the circle.

• Finding Terms in a Binomial Expansion [Jen, 03/27/2001]
  How can I find the fifth term of (-5m+4n^3)^54?

• Big O Notation and Polynomials [Paul, 04/12/2001]
  Given the function f(x) = (x^3 - (4x^2) + 12)/(x^2 + 2), how can I find a polynomial function g(x) such that f(x) = O(g(x)) and g(x) = O(f(x))?

• Using Cardan's Formula to find Real Roots [Salman, 04/13/2000]
  How can I show that, although the cubic equation x^3 - 6x = 4 has three real solutions, Cardan's formula can find them by subtracting appropriate cube roots of complex numbers?

• Coefficients in a Trinomial Expansion [Birkenfeld, 04/24/2001]
  In the expansion of (a+b+c)^6, what is the coefficient of a^2b^2c^2?

• Quadratic Function [Trey, 05/04/2001]
  What are the effects of changing the values of a, b, and c in a quadratic function?

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

• Irreducible Polynomials [McLoone, 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').

• Simplifying Radicals [Alec, 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?

• Are All Functions Equations? [Maureen, 07/16/2001]
  When my x's are not continuous, would I still have a function since the vertical line test might in fact not touch a point at all?

• Polynomials of the Fifth Degree and Above [Peter, 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?

• Show that x = 2 [Andrew, 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.

• Solving Third-Degree Equations [Jan, 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.

• Finding Roots of Polynomials with Complex Numbers [Ed, 09/27/2001]
  I read in the archives that you can find the roots of 3rd or higher- degree polynomials with complex numbers...

• Factoring Polynomials Using Substitution [Scott, 09/26/2001]
  I can't get two questions on my own: x^2+10xy+25y^2-81z^2 and a^4-6a^ 2b^2-27b^4.

• Degree of Constant Function [Masha, 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.

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

• Finding Zeros of Polynomials [Yazmin, 11/13/2001]
  Can you explain how to find the zeros of polynomials and give me an example of an application?

• Cubic Equations in One Formula [Robert, 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?

• Polynomial Long Division [Rachel, 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?

• Solving a Quartic Polynomial using a Trig Substitution [Kevin, 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.

• Factoring Cubic Polynomials [Ash, 01/01/2002]
  I have a lot of trouble factoring cubic polynomials like 10^3-3x^2-x= f(x). What does it mean to factor a polynomial into "linear factors"?

• Rational Expressions [Ashley, 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?

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

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

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

• Rearranging a Polynomial [Amanda, 02/23/2002]
  How do you do this question: 2n^2 + y^2 = 66^2 ?

• Factoring Trinomials [John, 02/24/2002]
  My teacher gave me a method for factoring trinomials (ax^2 +bx +c). It's not FOIL or the grouping method, though the first part is similar. What's the logic/reason behind this method?

• Logarithms of the Zeros of a Quadratic [Mary, 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?