Calculus Solver Simplifying expressions
Simplifying means rewriting an expression so it is shorter but worth exactly the same. You expand brackets, combine like terms and cancel common factors. Nothing about the value may change: if it does, you have solved or broken something instead of simplifying.
The order that works
Expand brackets first, then combine like terms, then cancel. Doing it in that order avoids most of the trouble, because you cannot see which terms are alike while they are still trapped inside brackets.
Like terms are the ones with exactly the same variable part: 3x and 2x are alike, 3x and 3x² are not, and 3x and 3y are not. Only their coefficients get added.
The errors that change the value
The minus sign in front of a bracket is the classic. 5 - (x - 3) is 5 - x + 3, which is 8 - x. Distributing the minus to only the first term gives 2 - x, a different expression.
The other classic is cancelling a term instead of a factor. In (x + 3)/3 the 3 on top is added, not multiplied, so it cannot cancel: the answer is not x + 1. Cancelling is only allowed on factors of the whole top and bottom.
How to check yourself
Substitute a number. Pick something awkward like x = 4, evaluate the original and evaluate your answer. If the two disagree, the simplification is wrong, and you found out in ten seconds.
Avoid x = 0 and x = 1 for the check: they hide too many errors, because multiplying by one or zero forgives most mistakes.
Common simplifications
| Expression | Simplifies to | Step used |
|---|---|---|
| 4x + 3x | 7x | Combine like terms |
| 2(x + 5) | 2x + 10 | Expand |
| 5 - (x - 3) | 8 - x | Distribute the minus |
| 6x²/2x | 3x | Cancel a common factor |
| x + x² | x + x² | Nothing: the terms are unlike |
Questions
What are like terms?
Terms with the same variables raised to the same powers. 4x²y and 7x²y are like terms; 4x²y and 4xy² are not.
Can you cancel the 3 in (x + 3)/3?
No. The 3 on top is added to x, not multiplied by the whole numerator, so it is not a factor. Only factors cancel.
Is simplifying the same as solving?
No. Simplifying rewrites an expression without changing its value. Solving finds the value of the unknown that makes an equation true, and needs an equals sign to work with.
More calculus reference
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