Algebra Calculator & Algebra Solver with Steps (Free)

An algebra calculator, also called an algebra solver, simplifies expressions, factors polynomials and solves equations step by step, keeping every answer as an exact fraction or surd.

Example: 2(3x − 5) = 4x + 6. Expand to 6x − 10 = 4x + 6, move terms to get 2x = 16, then divide by 2: x = 8.

Rule of thumb: always check by substituting back. With x = 8, the left side is 2(24 − 5) = 38 and the right side is 32 + 6 = 38, so the answer is correct.

An algebra calculator is a tool that works with letters as well as numbers. A normal calculator can only tell you that 3 × 4 = 12. An algebra calculator can tell you that 3(x + 4) = 3x + 12, that x² − 9 = (x − 3)(x + 3), and that 3x + 12 = 0 when x = −4.

Many students search for it as an algebra solver: the name is different, but the job is the same. You type the problem, and the solver finds the unknown and shows the working.

What it is used for: checking homework, building answer keys, learning the method from the steps, and getting an exact result fast. The letters stand for unknown numbers. The job of algebra is to find them, or to rewrite an expression in a simpler or more useful shape.

A worked one-liner: for 2(3x − 5) = 4x + 6, the calculator expands the bracket to 6x − 10, collects the x terms to get 2x = 16, divides to get x = 8, and then checks that both sides equal 38.

Who uses it: pre-algebra students (Grades 6 to 8) checking fractions and one-step equations; Algebra 1 students (Grades 8 and 9) solving linear equations, inequalities and systems; Algebra 2 students (Grades 10 and 11) factoring polynomials and solving quadratics with complex roots; IGCSE, GCSE, CBSE, SAT and ACT candidates; first-year university students in linear algebra; tutors and teachers building answer keys; and parents checking homework.

4.1 What the answer means

A solution is a number that makes the equation true. For 2(3x − 5) = 4x + 6 the answer x = 8 means that if you put 8 in place of x, both sides give the same number, 38. Any other number gives two different sides. A simplified expression is different: it is not a number at all, but a shorter way of writing the same thing. 3(x + 2) − 2(x − 4) and x + 14 give the same value for every x you try.

4.2 What the solution type tells you

A linear equation (highest power 1) has exactly one solution, no solution, or every number as a solution. If the x terms cancel and leave something false like 6 = 5, there is no solution. If they cancel and leave something true like 6 = 6, the equation is an identity and every x works. A quadratic (highest power 2) can have two different roots, one repeated root, or two complex roots. The discriminant b² − 4ac decides which.

4.3 Exact answers versus decimals

The calculator keeps every number exact. It turns 0.75 into 34 before it starts, so 0.1 + 0.2 gives exactly 310, not 0.30000000000000004. Roots that are not whole numbers stay as surds, for example 3 ± √174. Teachers usually want the exact form. Use the decimal only when the question says “to 2 decimal places” or when the number is a measurement.

4.4 How to check the answer yourself

Substitute the answer into the original equation, not into one of your working lines, because a mistake in an early line would carry through. For a factorisation, multiply the brackets back out. For a simplified expression, put in a simple number like x = 2 in both the original and the answer. If the two values match, you are almost certainly right. The green Verified badge means the calculator has done this check exactly.

4.5 What the graphs show

Chart 1 draws the expression as a curve. Its x-intercepts (where the curve crosses the horizontal axis) are the roots. Chart 2 draws the left side and the right side as two separate curves; the solution is where they meet. Chart 3 places every root on the complex plane: real roots sit on the horizontal axis, and complex roots appear as a mirror pair above and below it. Chart 4 shows the size and sign of each coefficient.

4.6 Excluded values and extraneous roots

When an equation has x in a denominator, some values are banned because they would divide by zero. In xx − 2, x cannot be 2. If clearing the fractions produces x = 2 as an answer, that root is extraneous: it came from the algebra, not from the original equation. The calculator lists excluded values first and crosses out any extraneous root.

4.7 How the answer changes if you change the numbers

For ax + b = c the solution is x = (c − b) ÷ a. Raising c by 1 raises x by 1 ÷ a. So when a is small, x is very sensitive to the other numbers, and when a is 0 the equation stops being solvable for x at all. For a quadratic, a small change in c moves the parabola up or down. That can turn two roots into one repeated root and then into a complex pair.

4.8 Where this leads next

Linear equations lead to systems of equations and to graphs of straight lines. Factoring leads to solving quadratics, cancelling algebraic fractions and finding the zeros of polynomials. Matrix rank leads into linear algebra. If your answer here came out as a quadratic, the natural next step is the quadratic formula; if it came out as two unknowns, the next step is a system of equations.

▶ Solve a problem above to fill all five cards with your own answer.

This flowchart shows how the algebra solver in this calculator works through a linear equation step by step, from the raw problem to a checked answer. Every box uses the same worked example as the Excel, R, Python and by-hand sections: 2(3x − 5) = 4x + 6.

Check any box yourself: the Excel sheet in Section 9, the R script in Section 10 and the Python script in Section 11 all return x = 8 with both sides equal to 38. Section 14 shows the same working by hand.

1. Type your problem or pick a sample. Enter an expression, equation, inequality, system or matrix in the box, or choose one of the 10 sample problems.
2. Use the keypad on a phone. The keys add x, y, powers, brackets, the = sign and the inequality signs without hunting for symbols.
3. Choose what to do. Leave the mode on Auto detect, or pick Simplify, Factor, Solve, Inequality, System, Standard form, Evaluate or Matrix rank.
4. Set the letter to solve for. If the equation has more than one letter, type the one you want in the Solve for box, for example x in ax + b = c.
5. Add values to evaluate. For Evaluate mode, type values like x = 2, y = -1 in the Evaluate at box.
6. Choose decimal places and answer form. Pick 2, 4, 6 or 10 decimal places, and exact and decimal or decimal only.
7. Press Solve. Nothing is calculated until you press Solve or Enter. Changing the sample clears the old results.
8. Read the answer and the steps. The green box gives the exact answer first, then the decimal, the solution type and a Verified badge.
9. Check the four charts. They show the graph, both sides of the equation, the roots and the coefficients.
10. Copy or download the working. Section 5 gives five ready-to-copy answer formats, and Download Doc or Download PDF saves every step.

Excel has no “solve for x” button and no symbolic algebra, so it cannot simplify or factor. It can still solve equations in three reliable ways: write the two sides as formulas and use Goal Seek, use the coefficient formula for a linear equation, or type the quadratic formula. Matrix functions solve systems. Every example uses the headline equation 2(3x − 5) = 4x + 6, and every route gives x = 8.

9.1 The functions you need

9.2 Step by step

9.3 The awkward cases

Complex roots. SQRT of a negative number gives #NUM!. When b² − 4ac is negative, use =IMSQRT(B3^2-4*B2*B4) and build the roots with IMSUM and IMDIV, or simply report that there are no real roots.

Goal Seek finds only one root. A quadratic has two. Run Goal Seek twice from two different starting values of x, one on each side of the vertex, to find both.

Rounding. Goal Seek stops when the difference is very small, not exactly 0, so it may show 7.99999. Round with =ROUND(A2,6) or use the exact formula in step 3.

Array formulas. MMULT and MINVERSE return several cells. Older Excel needs Ctrl+Shift+Enter; Excel 365 spills the answer down by itself and shows #SPILL! if a cell below is not empty.

9.4 Why Excel might disagree with R, Python or your textbook

9.5 Charting it in Excel

To see the solution, fill A2:A12 with x = 0 to 10, put the two sides in B and C, select all three columns and insert a Scatter with Straight Lines chart. Never use a Line chart: it spaces the x values evenly whatever they are. The lines cross at (8, 38). Two fixes help: set the horizontal axis bounds to 0 and 10, and add data labels to the crossing point only.

9.6 Excel errors you will probably hit

Base R solves linear and polynomial equations numerically and systems exactly enough for most work, with no packages to install. The script solves the headline equation 2(3x − 5) = 4x + 6, checks it, then shows a quadratic, a system and a matrix rank, and saves a 300 dpi figure of both sides meeting at x = 8.

10.1 The complete script

R, base only, saves algebra_solution.png

# ============================================================
# Algebra calculator in R: solve, check, quadratic, system, rank
# StatsUnlock - https://statsunlock.com/algebra-calculator/
# Base R only. No packages to install.
# ============================================================

# ---- 1. The equation: 2(3x - 5) = 4x + 6 -------------------
lhs <- function(x) 2 * (3 * x - 5)
rhs <- function(x) 4 * x + 6
f   <- function(x) lhs(x) - rhs(x)      # solve f(x) = 0

# ---- 2. Linear shortcut: intercept and slope of f -----------
b0 <- f(0)                  # -16
b1 <- f(1) - f(0)           # 2
x  <- -b0 / b1              # 8
cat("Solution: x =", x, "\n")

# ---- 3. Check by substitution -------------------------------
cat("Check: LHS =", lhs(x), " RHS =", rhs(x), "\n")
if (isTRUE(all.equal(lhs(x), rhs(x)))) cat("Verified: both sides are equal.\n")

# ---- 4. Same answer with a numeric root finder --------------
r <- uniroot(f, c(-100, 100), tol = 1e-10)
cat("uniroot gives x =", round(r$root, 6), "\n")

# ---- 5. Quadratic 2x^2 - 3x - 1 = 0 -------------------------
# polyroot wants ascending order: c, b, a
roots <- polyroot(c(-1, -3, 2))
print(sort(round(Re(roots), 4)))     # -0.2808  1.7808

# ---- 6. System: 2x + 3y = 12, x - y = 1 ---------------------
A <- matrix(c(2,  3,
              1, -1), nrow = 2, byrow = TRUE)
b <- c(12, 1)
print(solve(A, b))                   # 3 2

# ---- 7. Matrix rank -----------------------------------------
M <- matrix(c(1, 2, 3,
              2, 4, 6,
              1, 0, 1), nrow = 3, byrow = TRUE)
cat("rank =", qr(M)$rank, "\n")      # 2

# ---- 8. Fractions in the output (MASS ships with R) ---------
A2 <- matrix(c(3, 2,
               1, 4), nrow = 2, byrow = TRUE)
print(MASS::fractions(solve(A2, c(1, 2))))   # 0 1/2

# ---- 9. Figure: both sides as lines, meeting at x = 8 -------
png("algebra_solution.png", width = 2000, height = 1400, res = 300)
xs <- seq(0, 10, by = 0.1)
plot(xs, lhs(xs), type = "l", lwd = 2.5, col = "#2563eb",
     xlab = "x", ylab = "value of each side",
     main = "2(3x - 5) = 4x + 6", ylim = c(-12, 52))
lines(xs, rhs(xs), lwd = 2.5, col = "#ea580c")
abline(h = 0, col = "#475569")
points(x, lhs(x), pch = 19, cex = 1.6, col = "#16a34a")
text(x, lhs(x), labels = paste0("(", x, ", ", lhs(x), ")"), pos = 2)
legend("topleft", c("LHS = 2(3x - 5)", "RHS = 4x + 6"),
       col = c("#2563eb", "#ea580c"), lwd = 2.5, bty = "n")
dev.off()

10.2 What each part does

10.3 What the figure shows

The blue line is the left side 2(3x − 5) and the orange line is the right side 4x + 6. They cross at the green point (8, 38). The x value of the crossing is the solution, and the y value is the number both sides equal, which is the check.

10.4 Common R problems and fixes

10.5 Useful one-liners

Python with SymPy does real symbolic algebra: it solves, expands, factors and simplifies with exact fractions and surds, the same way this calculator does. The script solves the headline equation, checks it, repeats every sample type from Section 2, and saves a 300 dpi figure.

Install once with pip install sympy numpy matplotlib. Google Colab already has all three.

11.1 The complete script

Python 3, SymPy, saves algebra_solution.png

# ============================================================
# Algebra calculator in Python: solve, simplify, factor, systems
# StatsUnlock - https://statsunlock.com/algebra-calculator/
# pip install sympy numpy matplotlib
# ============================================================
from sympy import symbols, Eq, solve, expand, factor, together, Rational, Matrix, linsolve
import numpy as np
import matplotlib
matplotlib.use("Agg")          # save to file; remove this line in Jupyter
import matplotlib.pyplot as plt

x, y = symbols("x y")

# ---- 1. The headline equation: 2(3x - 5) = 4x + 6 ----------
eq = Eq(2*(3*x - 5), 4*x + 6)
sol = solve(eq, x)
print("Solution:", sol)                                   # [8]

# ---- 2. Check by substitution -------------------------------
x0 = sol[0]
lhs, rhs = eq.lhs.subs(x, x0), eq.rhs.subs(x, x0)
print("Check: LHS =", lhs, " RHS =", rhs)                 # 38 and 38
assert lhs == rhs
print("Verified: both sides are equal.")

# ---- 3. Simplify (expand and combine like terms) ------------
print(expand(3*(x + 2) - 2*(x - 4) + x**2))              # x**2 + x + 14

# ---- 4. Factor ----------------------------------------------
print(factor(6*x**2 + 11*x - 35))                         # (2*x + 7)*(3*x - 5)

# ---- 5. Quadratic with exact surds --------------------------
print(solve(Eq(x**2/2 - Rational(3, 4)*x, Rational(1, 4)), x))  # [3/4 - sqrt(17)/4, 3/4 + sqrt(17)/4]

# ---- 6. Algebraic fractions ---------------------------------
print(together(1/x + 1/(x + 2)))                          # 2*(x + 1)/(x*(x + 2))

# ---- 7. System of equations ---------------------------------
print(linsolve([2*x + 3*y - 12, x - y - 1], [x, y]))      # {(3, 2)}

# ---- 8. Matrix rank -----------------------------------------
M = Matrix([[1, 2, 3], [2, 4, 6], [1, 0, 1]])
print("rank =", M.rank())                                 # 2

# ---- 9. Decimals become exact fractions first ---------------
d = solve(Eq(Rational("0.5")*x + Rational("1.25"), Rational("2.75") - Rational("0.25")*x), x)
print("Decimal equation:", d)                             # [2]

# ---- 10. Figure: both sides as lines, meeting at x = 8 ------
xs = np.linspace(0, 10, 200)
fig, ax = plt.subplots(figsize=(6.5, 4.5))
ax.plot(xs, 2*(3*xs - 5), color="#2563eb", lw=2.5, label="LHS = 2(3x - 5)")
ax.plot(xs, 4*xs + 6, color="#ea580c", lw=2.5, label="RHS = 4x + 6")
ax.scatter([8], [38], s=80, color="#16a34a", zorder=3)
ax.annotate("(8, 38)", (8, 38), textcoords="offset points", xytext=(-52, 6))
ax.axhline(0, color="#475569", lw=1)
ax.set(xlabel="x", ylabel="value of each side", title="2(3x - 5) = 4x + 6")
ax.legend(frameon=False)
fig.savefig("algebra_solution.png", dpi=300, bbox_inches="tight")
print("Saved algebra_solution.png")

11.2 What each part does

11.3 What the figure shows

The same picture as the R figure: the two sides as straight lines meeting at (8, 38). If you change the equation to one with no solution, such as 2(x + 3) = 2x + 5, the two lines become parallel and never meet.

11.4 Common Python problems and fixes

11.5 Useful one-liners

Run it from a terminal with python algebra.py, in Jupyter (add plt.show() to see the figure inline), or paste it into Google Colab.

12.1 Properties of algebra

How to use: every step of a solution is one of these rules. Naming the rule beside each line is what earns method marks.

12.2 Exponent rules

12.3 Special products and factoring patterns

How to use: read across from the left column. Before any pattern, always take out the greatest common factor.

12.4 Discriminant cases for ax² + bx + c = 0

12.5 Inequality rules and interval notation

12.6 Perfect squares and cubes from 1 to 15

How to use: spotting these numbers is how you recognise a difference of squares, a perfect square trinomial or a sum or difference of cubes, and how you simplify a surd: √48 = √(16 × 3) = 4√3.

12.7 How to type maths into this calculator

12.8 Common algebra problems with answers

The problems students search for most, each solved exactly. Tap Steps to open the full step-by-step working in the calculator above.

Linear equations

Quadratic equations

Simplify and expand

Factoring

Inequalities

Systems and lines

This is the pen and paper method a teacher expects to see in an exam, worked on the same example as Sections 7, 9, 10 and 11: 2(3x − 5) = 4x + 6.

14.1 The written method

1. Copy the equation exactly. 2(3x − 5) = 4x + 6. Most lost marks start with a miscopied sign.
2. Expand the brackets. Multiply the 2 by both terms inside: 2 × 3x = 6x and 2 × (−5) = −10. Now 6x − 10 = 4x + 6.
3. Collect the x terms on one side. Subtract 4x from both sides: 2x − 10 = 6. Choose the side that keeps the x coefficient positive.
4. Collect the numbers on the other side. Add 10 to both sides: 2x = 16.
5. Divide by the coefficient of x. x = 16 ÷ 2 = 8.
6. Check in the original equation. Left: 2(3 × 8 − 5) = 2 × 19 = 38. Right: 4 × 8 + 6 = 38. Both sides agree.
7. Write the answer clearly. x = 8, or as a solution set {8}.

For a quadratic, the method changes after step 4: get everything on one side so it equals 0, take out any common factor, try to factor (two numbers that multiply to ac and add to b), and use the quadratic formula if it will not factor. Never divide both sides by x.

14.2 On a scientific calculator

The basic Casio fx-82MS and TI-30XIIS have no SOLVE function; work the steps by hand and use the calculator only for the arithmetic. Keystrokes differ slightly between models and software versions, so confirm them in your calculator's manual. SOLVE finds one root near the start value, so for a quadratic run it twice from different start values.

14.3 Practice problems with answers

14.4 Mistakes that lose marks

• Multiplying only the first term inside a bracket: −2(x − 4) is −2x + 8, not −2x − 4 or −2x − 8.
• Moving a term across the = sign without changing its sign.
• Forgetting to flip an inequality when dividing by a negative number.
• Dividing both sides by x, which throws away the root x = 0.
• Cancelling terms instead of factors: x + 33 is not x.
• Rounding a surd too early, which turns an exact answer into a slightly wrong decimal.
• Keeping an extraneous root that makes a denominator zero.
• Not checking. Substitution takes thirty seconds and catches nearly every slip.

The right method depends on what the problem looks like. Picking well saves time and avoids messy arithmetic.

Use this algebra solver when:

• ✓ You want to check homework and see exactly which step went wrong.
• ✓ You need an exact answer with fractions or surds, not a rounded decimal.
• ✓ You are building an answer key for many problems (Batch tab).
• ✓ You want a graph that shows why the answer is right.
• ✓ The problem is a polynomial, an algebraic fraction, a linear system or a matrix.

Use something else when:

• ✗ The equation has sin, log, e^x or √x: use a scientific or graphing calculator.
• ✗ You need calculus (derivatives, integrals, limits).
• ✗ You need to factor a polynomial with three or more letters that is not homogeneous or groupable.
• ✗ The exam forbids calculators: practise with Section 14 until the steps are automatic.

Four real examples:

1. Phone plans: plan A costs 20 + 0.05m and plan B costs 35 + 0.02m. Solve 20 + 0.05m = 35 + 0.02m to find the break-even, m = 500 minutes.
2. Garden design: a rectangle has length 3 m more than its width and area 40 m². Solve w(w + 3) = 40 to get w = 5 m.
3. Ticket sales: 250 tickets at 12 and 7 dollars raised 2500 dollars. A 2 by 2 system gives 150 adult and 100 child tickets.
4. Physics: a ball's height is h = −5t² + 20t. Solve −5t² + 20t = 0 to find it lands at t = 4 seconds.

Decision tree

Assumptions

1. Every letter is a variable. xy means x × y. Check: a word like “area” would be read as a × r × e × a. Use single letters.
2. Numbers are exact. 0.333 is treated as 3331000, not 13. Type 1/3 when you mean a third.
3. Roots are reported over the complex numbers. Real roots come first; complex roots are listed when the discriminant is negative. Inequalities use real numbers only.
4. The domain excludes zeros of every denominator in the original problem, even when a factor later cancels.
5. Exponents are whole numbers. x^-1 is fine and means 1/x; x^(1/2) is a root and is not simplified here.

Limitations

1. No functions: sin, cos, log, ln, e^x and √x are not handled. Use a scientific or graphing calculator.
2. Polynomials of degree 5 or more usually have no formula for their roots (Abel-Ruffini theorem). The calculator finds rational roots exactly and the rest numerically.
3. Factoring with three or more letters covers the common factor, grouping and homogeneous two-letter cases, not every multivariable polynomial.
4. Equations with two letters are solved for one letter only when they are linear in that letter.
5. Systems must be linear (no x², no xy). Non-linear systems need substitution by hand or a computer algebra system.
6. Size limits: powers up to 60, matrices up to 8 by 8, and polynomials up to 3000 terms after expanding.

How this calculator was tested

• Exact engine: every number is a fraction of two whole numbers (BigInt), so there is no rounding until the final decimal line.
• 45 engine test cases covering every mode, error message and edge case (repeated roots, complex roots, extraneous roots, singular systems), plus the 35 problems in Section 12.8, all giving the expected answer.
• Cross-checked against SymPy in Python and base R: the scripts in Sections 10 and 11 print the same answers as this page.
• Self-check on every answer: the calculator substitutes its own answer back into your original problem before showing the Verified badge.

What the answer is

This algebra calculator gives an exact answer first, as a whole number, a fraction or a surd, then the decimal. For the headline problem 2(3x − 5) = 4x + 6, that answer is x = 8, and the check shows both sides equal 38.

What it tells you

The solution type matters as much as the number. One solution, no solution, every number, a repeated root or a complex pair each say something different about the problem. On a graph they are a crossing, two parallel lines, two identical lines, a curve touching the axis, or a curve that never reaches it.

Watch out for

Sign errors when terms cross the = sign, forgetting to flip an inequality, dividing by x, and extraneous roots from denominators cause most wrong answers. The step list shows each move on its own line, so you can find the exact line where your working first differs.

Next steps

If your equation turned out to be a quadratic, practise factoring and the quadratic formula. If it has two unknowns, move to systems of equations. For number work, the factor calculator, the mixed number calculator and the proportion calculator cover the arithmetic that algebra is built on. Whatever the problem, the algebra solver works the same way: exact answer, every step, and a check.

Teachers and tutors: paste the embed code into a class page or blog. The embedded version shows only the calculator and the results.

These peer-reviewed articles support the methods behind this algebra calculator and algebra solver: how students learn to solve equations and where they go wrong, exact polynomial arithmetic and factoring, numerical root finding, and the use of computer algebra in teaching.

1. Booth, J. L., Barbieri, C., Eyer, F., & Paré-Blagoev, E. J. (2014). Persistent and pernicious errors in algebraic problem solving. Journal of Problem Solving, 7(1), Article 3. https://doi.org/10.7771/1932-6246.1161
2. Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27(1), 59–78. https://doi.org/10.1007/BF01284528
3. Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326. https://doi.org/10.1007/BF00311062
4. Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30(1), 39–65. https://doi.org/10.1007/BF00163752
5. MacGregor, M., & Stacey, K. (1997). Students' understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1–19. https://doi.org/10.1023/A:1002970913563
6. Stacey, K., & MacGregor, M. (1999). Learning the algebraic method of solving problems. Journal of Mathematical Behavior, 18(2), 149–167. https://doi.org/10.1016/S0732-3123(99)00026-7
7. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574. https://doi.org/10.1037/0022-0663.99.3.561
8. Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565–579. https://doi.org/10.1016/j.learninstruc.2007.09.018
9. Tall, D., de Lima, R. N., & Healy, L. (2014). Evolving a three-world framework for solving algebraic equations in the light of what a student has met before. Journal of Mathematical Behavior, 34, 1–13. https://doi.org/10.1016/j.jmathb.2013.12.003
10. Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction on students' understanding of quadratic equations. Mathematics Education Research Journal, 18(1), 47–77. https://doi.org/10.1007/BF03217429
11. Kieran, C., & Drijvers, P. (2006). The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning, 11(2), 205–263. https://doi.org/10.1007/s10758-006-0006-7
12. Pierce, R., & Stacey, K. (2004). A framework for monitoring progress and planning teaching towards the effective use of computer algebra systems. International Journal of Computers for Mathematical Learning, 9(1), 59–93. https://doi.org/10.1023/B:IJCO.0000038246.98119.14
13. Collins, G. E. (1967). Subresultants and reduced polynomial remainder sequences. Journal of the ACM, 14(1), 128–142. https://doi.org/10.1145/321371.321381
14. Zassenhaus, H. (1969). On Hensel factorization, I. Journal of Number Theory, 1(3), 291–311. https://doi.org/10.1016/0022-314X(69)90047-X
15. Lenstra, A. K., Lenstra, H. W., & Lovász, L. (1982). Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4), 515–534. https://doi.org/10.1007/BF01457454
16. Bareiss, E. H. (1968). Sylvester's identity and multistep integer-preserving Gaussian elimination. Mathematics of Computation, 22(103), 565–578. https://doi.org/10.1090/S0025-5718-1968-0226829-0
17. Kerner, I. O. (1966). Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen. Numerische Mathematik, 8(3), 290–294. https://doi.org/10.1007/BF02162564
18. Aberth, O. (1973). Iteration methods for finding all zeros of a polynomial simultaneously. Mathematics of Computation, 27(122), 339–344. https://doi.org/10.1090/S0025-5718-1973-0329236-7
19. Jenkins, M. A., & Traub, J. F. (1970). A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration. Numerische Mathematik, 14(3), 252–263. https://doi.org/10.1007/BF02163334
20. Goldberg, D. (1991). What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys, 23(1), 5–48. https://doi.org/10.1145/103162.103163