Algebra turns real situations into symbols, but raw expressions are often messy: half a dozen terms scattered across parentheses, constants you have to combine in your head, and products you were never told to multiply out. This simplify expression calculator handles the two moves students ask for most: combining like terms in a linear expression, and expanding a product of two binomials. You type the coefficients instead of the string, and the engine returns the exact coefficient of x, the coefficient of x squared, and the constant term. It is a fast, trustworthy shortcut for the problems that actually show up on homework sheets.
For a linear sum like 3x + 5x - 2x, set the mode to 1 and enter 3, 5, and -2. The result card gives a coefficient of 6, which means the whole thing collapses to 6x. For a product like (x + 2)(x - 3), switch the mode to 2 and enter 2 and -3. The card gives a coefficient of x squared of 1, a coefficient of x of -1, and a constant of -6, which is exactly x^2 - x - 6. No guessing, no rounding.
A quick honesty note: this page is not a full computer algebra system. It cannot parse a typed string, factor an arbitrary quadratic, or simplify a rational expression. Those jobs need a tool such as SymPy. It is a combine like terms calculator and a expand and simplify calculator for the two cases above, and it does those cases exactly.
How to Use This Calculator
- Pick a Mode. Use 1 for combine like terms (several x terms added together). Use 2 for expanding a product
(x + p)(x + q). - In mode 1, enter the three x-coefficients in boxes a, b, and c. Enter the plain number in box d. If you only have two x terms, set the third to 0.
- In mode 2, enter the two binomial constants p and q in boxes a and b. Boxes c and d are optional and are ignored in this mode — leave them at 0.
- Read the result card from top to bottom: Coefficient of x squared, Coefficient of x, and Constant term. The section below reassembles them into the finished expression.
- Use the Result evaluated at x = 1 line as a quick sanity check: plug x = 1 into both your original expression and the simplified one, and the two numbers must match.
How to Read the Result as a Finished Expression
The card prints three coefficients, and you reassemble the expression from them in standard order — highest power first:
- Mode 1 result reads
(Coefficient of x)x + (Constant term). For example, coefficient of x = 6 and constant = 0 gives6x; coefficient of x = 7 and constant = 7 gives7x + 7. If the x coefficient is 0, the whole x part vanishes and only the constant remains. - Mode 2 result reads
(Coefficient of x squared)x² + (Coefficient of x)x + (Constant term). For example, x squared = 1, x = −1, constant = −6 givesx² − x − 6. If the x coefficient is 0 the middle term vanishes (as inx² − 25), and a positive constant means you add it instead of subtracting.
The Formula Behind It
Combining like terms. Every term that carries the same variable part is added. If you start from ax + bx + cx + d, the x terms become a single coefficient:
combined = (a + b + c) x + d
For example 3x + 5x - 2x + 7 collapses to (3 + 5 - 2)x + 7 = 6x + 7. The constant d is carried untouched.
Expanding a binomial. A product (x + p)(x + q) multiplies out by the distributive property:
(x + p)(x + q) = x^2 + (p + q) x + p * q
The coefficient of x squared is always 1 (because both factors start with x). The coefficient of x is the sum p + q, and the constant is the product p * q. Two common shortcuts fall straight out of this formula:
- Difference of squares. When q = -p, the linear term cancels and you get
(x + p)(x - p) = x^2 - p^2. - Perfect square. When p = q, you get
(x + p)^2 = x^2 + 2p x + p^2.
Reference: the distributive property and FOIL expansion are standard high-school algebra; see the SymPy simplification docs for what a full symbolic engine can do beyond these two cases.
Examples
Example 1 — Combine like terms. You buy 3 notebooks that cost x dollars, then 5 more, and return 2. The bill is 3x + 5x - 2x. Set mode 1, enter a = 3, b = 5, c = -2, d = 0. The card shows coefficient of x = 6, constant = 0, so the simplified total is 6x. If a notebook is $4, the total is 6 × $4 = $24.
Example 2 — Expand a rectangle. A rectangle has sides (x + 2) and (x - 3). Set mode 2, enter a = 2, b = -3. The card shows x squared = 1, x = -1, constant = -6, so the area polynomial is x^2 - x - 6. At x = 5 the area is 25 - 5 - 6 = 14, which you can verify directly from 7 × 2 = 14.
Example 3 — Difference of squares. A square of side (x + 5) minus a square of side (x - 5) is often written as the product (x + 5)(x - 5). Set mode 2, enter a = 5, b = -5. The card shows x squared = 1, x = 0, constant = -25, so the expression simplifies to x^2 - 25. The linear term vanishes exactly.
Example 4 — Combine with a constant. Your expression is 4x + 2x + x + 7. Set mode 1, enter a = 4, b = 2, c = 1, d = 7. The card shows coefficient of x = 7, constant = 7, giving 7x + 7. Evaluated at x = 1 that is 14, which matches the original.
FAQ
What does it mean to simplify an algebraic expression?
Simplifying means rewriting an expression in its shortest standard form by combining like terms and reducing constants. 3x + 5x - 2x becomes 6x, and (x + 2)(x - 3) expands to x^2 - x - 6. The value of the expression never changes; only its appearance does.
How do I combine like terms?
Add the coefficients of terms that share the same variable part. In 7y + 2y - 4y, all three terms are in y, so 7 + 2 - 4 = 5, giving 5y. Terms with different variables, such as 3x and 4y, cannot be combined with each other.
How do I expand (x + p)(x + q)?
Use x^2 + (p + q)x + p * q. For (x + 2)(x - 3) the coefficient of x is 2 + (-3) = -1 and the constant is 2 * (-3) = -6, so the result is x^2 - x - 6.
What is the difference between simplifying and factoring?
Simplifying here means collapsing a sum of like terms or expanding a product into standard polynomial form. Factoring goes the other way: it takes x^2 - x - 6 back to (x + 2)(x - 3). This calculator does the forward move (expand and combine); it does not factor for you.
Can this calculator handle y, z, or other variables?
Yes, as long as all your terms use the same variable. The variable letter itself does not affect the arithmetic: 3y + 5y - 2y combines the same way as 3x + 5x - 2x, giving 6y. If your expression mixes variables, combine each variable group separately by running the calculator once per group.
What kinds of expressions can this tool not simplify?
It cannot parse a typed string, factor a quadratic for you, cancel common factors inside a fraction, or handle powers higher than x squared in a binomial product. For those, use a full computer algebra system such as SymPy. This page is intentionally limited to the two closed-form shortcuts above so that every answer is exact.
How do I check that the result is right?
Plug x = 1 into both the original expression and the simplified one. The calculator shows this number on the Result evaluated at x = 1 line. If the two values agree, the coefficients are correct.
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Disclaimer
This simplify expression calculator is provided for informational purposes only and does not constitute mathematical, academic, or professional advice. Every result is produced by exact closed-form arithmetic for the two supported shortcuts; for any other expression, verify the result by hand or with a qualified teacher. Always double-check final work before using it in homework, engineering, finance, or any graded or published setting.