Radicals and rational expressions
Roots and fractions with polynomials in them both hide their simplest form. How do you rewrite them exactly, and solve equations that contain them without picking up false answers?
Where you are 5 inside
What this area is about
Two kinds of expression go past plain polynomials. A radical puts a quantity under a root, as in \displaystyle \sqrt{245}; a rational expression divides one polynomial by another, as in \displaystyle \frac{x + 1}{x^2 - 4}. Both are rewritten exactly, never rounded, and both carry restrictions: for a real value, a square root needs a quantity under it that is not negative, and a value that makes a denominator 0 is excluded. Solving equations with either can produce an answer that fails the original equation, so every answer is checked.
How these fit together
Radicals. Simplifying radicals pulls perfect squares out from under the root, combines roots and rationalizes denominators. Solving radical equations isolates the root and raises both sides to the matching power; squaring can bring in false answers.
Rational expressions. Simplifying rational expressions factors, cancels, and does the four operations while keeping excluded values excluded. Solving a rational equation multiplies through by the common denominator, then rejects any answer that makes a denominator 0.
A model. Inverse, joint and combined variation writes laws such as \displaystyle y = \frac{k}{x} with one constant k \ne 0.
A path through it
The two strands are separate. Radicals: simplifying, then radical equations. Rational expressions: simplifying, then rational equations and inverse variation.
After this: the quadratic formula, whose answers often leave a root, radical functions and rational functions.