Content of Chapter 5, 6, 7 and 8
Mathematical objects: Integer exponents, polynomials, rational expressions,
(principle) n-th roots, rational exponents, radical expressions, complex numbers,
quadratic equations, discriminant ∆.
Techniques:
1. Use rules of exponents to simply expressions (base must be positive in the
rational exponent case).
2. Simplify: rational expressions (common factor) and radical expressions
(three criteria of the simplest form).
3. +, −, × and ÷ between: polynomials (horizontal format and vertical
format, division with remainder(long division and synthetic division)),
rational expressions (complex fractions), radical expressions (conjugates)
and complex numbers (complex conjugates).
4. Factor polynomials (common monomial, by grouping, special products,
trinomials).
5. Solve polynomial (factoring), rational (numerator = 0), radical (isolate a
radical) and quadratic (completing the square, quadratic formula) equations.
6. Graph a quadratic function. Vertex and axis of parabola.
Things to notice:
1. While applying a formula such as quadratic formula or solving a polynomial equation by factoring, you should notice the formula or method can
ONLY be applied to a SPECIAL form of expressions or equations (e.g.
quadratic formula requires an equation looks like ax2 + bx + c = 0). You
should rewrite the expression of equation in that form first. Try to find all
techniques above who require a special form of expressions or equations.
2. Don’t forget the annoying absolute value when you want to apply the tautological properties of n-th root when the index is even. Discuss carefully
whether the absolute value can be removed.
Things won’t be covered in Midterm 2: Scientific notation, quadratic and
rational inequalities.