Most calculus textbooks have a review chapter before they begin exploring calculus. It is more effective to review the needed algebra as you go along. You'll see many complex algebraic topics come up in this story.
Algebra and Geometry needed for Part 1: Slopes & Velocities
Algebra
Determine the equation of a line given two points, or a point and a slope, or a graph of a line,
Find the average rate of change over an interval given a function or its graph,
Clearly express what is happening to an object given a position versus time graph,
Evaluate f(x+h) for any given function f(x),
Rationalize the numerator of expressions with square roots,
Simplify complex fractions (fractions within fractions).
Geometry (discussed in chapter 1)
Know the area and circumference of a circle,
Understand where perimeter, area, and volume ‘formulas’ come from.
Algebra with Calculus Concepts
Approximate, using two points close to each other, the instantaneous rate of change at a point, given a function or its graph,
Explain clearly why the procedure you used gives an approximation of the true instantaneous rate of change,
Sketch a velocity versus time graph given a position versus time graph,
Expand a parenthetical expression raised to a power,
Construct the formal definition of the derivative by modifying the definition of slope,
Apply the formal definition of the derivative to simple polynomials and to simple square root functions.
Algebra Skills needed for Part 2: So Many Derivatives
Algebra
Multiply out the expression (x+h)n (necessary to understand the proof for the derivative of y=xn),
Identify the holes, vertical asymptotes, x- and y-intercepts, horizontal or slant asymptote, and domain of any rational function,
Sketch the basic shape of a rational function,
Identify an equation for a rational function given a sketch of the function,
Explain clearly what a hole and an asymptote are,
Construct the equation of a piecewise function given its graph,
Sketch the graph of a piecewise function given its equation,
Work with inequalities,
Give both triangle and circle definitions of sin x, cos x, and tan x, and explain how they’re related,
Evaluate sin x, cos x, and tan x at all multiples of π/6 and π/4, without a calculator,
Understand trigonometry identities, including and sin(x+h)=sin x cos h + sin h cos x,
Accurately graph y = sin x and y = cos x,
Understand composition of functions,
Use logarithm properties to “break apart” a single logarithmic expression into simple logarithms,
Understand properties of exponents,
Be able to graph exponential and logarithmic functions.
Geometry
Understand where perimeter, area, and volume ‘formulas’ come from.
Algebra with Calculus Concepts
Graph a polynomial or rational function, showing its maximums, minimums, and inflection points,
Follow complicated logic (in the definition of limit),
Think in terms of nested functions to determine outer and inner functions, in order to use the chain rule.
Algebra Skills needed for Part 3: Anti-derivatives & Area
Algebra
Work with summations.