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High School Archive

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.

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Adding and Subtracting Polynomials [12/08/1996]

How do you add and subtract polynomials?

Binomial Expansions and Pascal's Triangle [7/10/1996]

What are binomial expansions and where can they be used?

Complex Roots [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? [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 [08/04/1998]

Some ideas for learning algebra that may help prevent frustration.

Factoring Polynomials [11/14/1996]

Is there a logic behind the factorization of polynomials?

Factoring Trinomials [04/29/2002]

Why and how we factor quadratic and other equations.

Finding Sum Formula using Sequences of Differences [06/28/1998]

Finding a formula for the sum of the first n fourth powers using sequences of differences.

How to Factor [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 [3/4/1995]

(x^4+4y^4)/(x^2-2xy+2y^2). Please show me carefully, step by step!

Multiplying Polynomials [07/10/2003]

Can you foil (x+2)(x+2)(x+2)?

Parent Pulling Her Hair Out [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 [09/11/2001]

Why are Pascal's triangle and the binomial coefficients the same?

Polynomial Basics and Terms [05/20/1998]

I'm having a lot of trouble understanding polynomials.

Polynomials: Terms, Exponents, Degrees [04/18/2002]

Can you give me an example of linear binomial, quadratic trinomial, etc.?

Regrouping Polynomials [11/19/2001]

I'm stuck on problems like a^2x - bx - a^2y + by + a^2z - bz.

Simplifying Algebraic Expressions [03/23/2002]

Please explain how to simplify algebraic expressions.

Synthetic Division [05/06/2003]

Divide a polynomial: (x^2-6x+9)/(x-3).

Adding Polynomials [02/15/1998]

I don't know how to add polynomials. Can you please help me?

Advanced Polynomial Factoring Methods [11/20/2005]

I've tried factoring x^5 + x + 1 using the Rational Roots Theorem, the quadratic formula, Descarte's rule of signs, common factors, and Newton's Method, but have had no luck. What else can I try?

Algebraic expressions and fractions [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)?

Applying the Remainder Theorem [11/19/2005]

Given f(x) = 7x^4 + 9x^3 + 4x^2 - 4x + 16, use the Remainder Theoreom to find f(2).

Are (-a)^3 and -a^3 the Same Thing? [11/08/2005]

One question on a test I took was (-a) * (-a) * (-a) = ?. I answered (-a)^3. My teacher said the answer is -a^3 (same answer without the parantheses). Aren't the two answers the same?

Are All Functions Equations? [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?

Avoiding the Final Step of Checking for Extraneous Solutions [01/10/2010]

A student seeks a technique that would let him skip the step of checking the roots of an equation in order to discard the extraneous solution. Doctor Peterson shows how to simplify that final check.

Big O Notation and Polynomials [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))?

Binomial Expansions [03/14/1999]

Find the constant term of the expansion of (x + x^-1)^6.

Binomial Theorem [8/5/1996]

Find the fourth term of (2a - 6b)^11.

Binomial Theorem by Induction [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.

Clearing Fractions in an Equation [01/13/2005]

How do I clear the fractions in 1/4(8y + 4) - 17 = -1/2(4y - 8)? I know I need to multiply by 4, but I get confused with all the parentheses involved in doing that.

Coefficients in a Trinomial Expansion [04/24/2001]

In the expansion of (a+b+c)^6, what is the coefficient of a^2b^2c^2?

Complete the Square and Factor [01/14/1997]

How do you complete the square? Can you use this to factor polynomials?

Completing the Cube [11/28/1996]

Is there some method analogous to completing the square for higher dimension polynomials?

Completing the Square: How Does it Work? [01/20/1999]

Can you explain why completing the square works? How exactly do you do it?

Complex Conjugate Roots of Real Polynomials [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?

Converting a Product Function to a Summation [07/24/2004]

How do I convert the product of n terms to a summation? For example, if f(x) = a(x - a1)(x - a2)...(x - an), how do I get f(x) = the sum of some series?

Counting Odd Coefficients [05/27/1998]

If (1+x)^100 is multiplied out, how many of the coefficients are odd? How would you generalize?

Cubic Convenience Factor [11/15/2010]

Given a linear equation in three variables, a student seeks help evaluating a cubic expression that features the same trio of unknowns. Doctor Ali provides a boost by using the distributive property to factor the cubic, then also suggests a clever substitution.

Cubic Curiosity [09/21/2013]

A student factors x^2 + x + 1 = 0, then substitutes a result back in for a term in the original equation, yielding a counterintuitive result. Doctor Peterson reveals the trouble with performing irreversible algebraic steps.

A Cubic Curve, Its Double Root, and Its Integer Solutions [02/27/2011]

A student seeks integer solutions to (x^2 + y)(x + y^2) = (x - y)^3. To oblige, Doctor Vogler parameterizes the equation of this twisted curve.

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