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Precalculus Review Problems

The algebra that quietly decides your AP Calc grade.

Ask any AP Calculus teacher where students lose points and you'll hear the same answer: not on the calculus. Students set up the derivative correctly and then drop a sign while distributing, mis-add fractions inside a difference quotient, or stall on a limit because they can't see the factoring. The calculus ideas are new; the algebra mistakes are old.

That's why a precalculus review is the highest-leverage prep you can do before (or during) AP Calc. Every technique below shows up inside calculus problems within the first month of the course:

Fraction arithmetic — difference quotients are stacked fractions.
Distribution and factoring (including special forms like the sum of cubes) — most 0/0 limits are solved by factoring and canceling.
Polynomial division and the remainder theorem — rewriting rational functions before differentiating or integrating.
Systems of equations — finding where curves intersect is step one of every area-between-curves problem.
Domains, holes, and asymptotes of rational functions — the language of limits and continuity.

Work the eight problems below without a calculator. Each answer comes with a worked solution, so a miss tells you exactly which skill to drill.

Practice problems

Problem 1

Evaluate

\frac{9}{7} - (- \frac{5}{9})

Problem 2

Write

2 (x^{2} - x - 8)

without parentheses

Problem 3

Factor

15 x^{2} + 8 x + 1

Problem 4

Factor

8 x^{3} + 125

Problem 5

Find the remainder when

p (x) = - 3 x^{3} - 3 x^{2} - 3 x + 7

is divided by

x + 2

Problem 6

Cost is

C (x) = 6 x + 12

and revenue is

R (x) = 7 x

. Find the break-even quantity

x

Problem 7

A function

f

has domain equal to all real numbers except

x = - 5

and

x = - 3

. Which interval notation describes this domain?

Problem 8

f (x) = \frac{( x - 4 ) ( x + 3 )}{x - 4}

Find the coordinates of the hole in the graph of

f

Want more than 8 problems?

These are fixed examples. Lodera is an adaptive question bank for AP Calculus AB and Precalculus that never runs out: miss a question and it serves you variations of the same skill until it sticks. Every problem is generated from a template and solved by a computer algebra system — exact answers, no AI-written solutions. 10 free questions a day, forever.

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