In Precalculus the cube patterns are a way to find every zero of a polynomial. The linear factor gives one real zero, and the quadratic factor, which never factors over the reals, gives two complex zeros through the quadratic formula.
Also taught in Algebra 2, at its own level and with its own examples.
Select your difficulty
Pick more than one for a sheet that mixes them.
The preview can look cramped on a phone. The PDF and printed sheet come out normal.
For x² + 2x + 4 = 0: x = (-2 ± √(4 - 16))/2 = (-2 ± 2i√3)/2.
The answer is x=2,−1±i3.
Where students go wrong
Stopping at x = 2. A cubic has three zeros, and the other two come from the quadratic factor even though it does not factor.
Also called the sum of cubes, the difference of cubes or cube factoring formulas.
What you can put on this worksheet
A difference of cubes
v3−64→ (v−4)(v2+4v+16)
A sum of cubes
v3+64→ (v+4)(v2−4v+16)
GCFs, constants first, two variables
27x3−64y3→ (3x−4y)(9x2+12xy+16y2)
Both terms negative, a GCF in two letters, solving
Solve: −27x3+125=0→ {35,6−5+5i3,6−5−5i3}
Questions about these worksheets
Yes. Every sum and difference of cubes sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the sum and difference of cubes questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 4: a difference of cubes, a sum of cubes, gcfs, constants first, two variables and both terms negative, a gcf in two letters, solving. Tick as many as you want and set how many of each, or let it spread them evenly.
Two things. The signs: in the second parenthesis the middle term always has the opposite sign to the first parenthesis, so a difference of cubes gives a plus in the middle and a sum gives a minus. And students try to keep factoring x² + 3x + 9. It never factors, on any of these, so the answer is finished.