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Understanding Basic Math – Test 2

 

Algebra of Polynomials

This is the second in a series of tests of understanding basic math. You need to be able to answer these questions accurately and confidently if you intend to take college-level math courses in Calculus, or Statistics. In answering the questions, please provide typical examples.

Exponents

  1. Explain how the expression xn is defined, if n is a positive whole number (n = 1,2,3,…) and x is any real number.
  2. Prove that xn xm  = xn+m.
  3. Prove that (xy)n  = xn yn and that (xn)m  = xnm.
  4. Explain why x0 is defined as x0  = 1 (for x not equal to 0).
  5. How is xn defined if n is a negative integer?
  6. Prove that xn / xm  = xn-m.

Polynomials

  1. Define “polynomial.” Also define “degree” of a polynomial.
  2. Show how to add and multiply two polynomials.
  3. Expand (x + a)2 and explain.
  4. Explain division of polynomials with remainder. Illustrate with an example.
  5. Show how to solve a linear equation ax + b = 0.

Quadratic Polynomials

  1. What is a quadratic polynomial?
  2. Explain how to factor a quadratic polynomial by trial and error.
  3. State the quadratic formula, with examples.
  4. Explain the method called “completing the square.”
  5. Use completing the square to prove the quadratic formula.
  6. Define the discriminant of a quadratic polynomial.

The Binomial Theorem

  1. Expand (a + b)3.
  2. Describe Pascal’s triangle.
  3. What are the binomial coefficients C(n,k)? How are they related to Pascal’s triangle?
  4. State the Binomial Theorem, using ∑ notation.

Roots of Polynomials

  1. Define the terms “root of a polynomial” and “factor of a polynomial.”
  2. Explain why any polynomial of odd degree must have at least one real root.
  3. State and prove the remainder theorem.
  4. State and prove the factor theorem.