## Understanding Basic Math – Test 2

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**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**

- Explain how the expression x
^{n}is defined, if n is a positive whole number (n = 1,2,3,…) and x is any real number. - Prove that x
^{n}x^{m}= x^{n+m}. - Prove that (xy)
^{n}= x^{n}y^{n}and that (x^{n})^{m}= x^{nm}. - Explain why x
^{0}is defined as x^{0}= 1 (for x not equal to 0). - How is x
^{n}defined if n is a negative integer? - Prove that x
^{n}/ x^{m}= x^{n-m}.

**Polynomials**

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

**Quadratic Polynomials**

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

**The Binomial Theorem**

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

**Roots of Polynomials**

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