Common Mistakes
Learn to recognize and avoid the most frequent errors students make with polynomials and radicals to improve your accuracy on tests and assignments.
Learn
Understanding common mistakes is just as important as learning the correct methods. This lesson highlights typical errors in polynomial operations and radical expressions so you can avoid them.
Why Study Mistakes?
- Errors often follow predictable patterns
- Recognizing mistakes helps you catch your own errors
- SAT and ACT test designers create wrong answers based on common errors
- Being aware of pitfalls improves your accuracy
Common Errors and Corrections
Study each mistake carefully, understand why it's wrong, and learn the correct approach.
Mistake 1: Squaring a Binomial Incorrectly
Why: When squaring a binomial, you must use (a + b)2 = a2 + 2ab + b2. The middle term (2ab) is often forgotten.
Mistake 2: Adding Radicals Incorrectly
Why: You cannot add radicands (the numbers under the radical). You must simplify each radical first, then add if possible.
Mistake 3: Distributing a Negative Sign Incorrectly
Why: When subtracting a polynomial, distribute the negative sign to ALL terms in the second polynomial.
Mistake 4: Simplifying Radicals Incompletely
Why: Always look for the largest perfect square factor to fully simplify. 36 is a larger perfect square factor of 72 than 4.
Mistake 5: Multiplying Exponents When Adding Polynomials
Why: When adding like terms, add the coefficients only. The variable and exponent stay the same.
Mistake 6: Canceling in Fractions with Radicals
Why: You can only cancel if you can divide ALL terms by the same factor. Simplify sqrt(8) = 2sqrt(2) first, then divide each term by 2.
Practice: Spot the Error
For each problem, identify the mistake and find the correct answer.
Problem 1
Student work: (2x + 5)2 = 4x2 + 25. Find the error.
Show Hint
What's missing from the expansion? Think about the pattern (a + b)2.
Show Answer
Missing the middle term 2(2x)(5) = 20x. Correct: 4x2 + 20x + 25
Problem 2
Student work: sqrt(16 + 9) = sqrt(16) + sqrt(9) = 4 + 3 = 7. Find the error.
Show Hint
Can you split a sum under a radical? Try computing sqrt(25).
Show Answer
sqrt(a + b) does not equal sqrt(a) + sqrt(b). Correct: sqrt(25) = 5
Problem 3
Student work: 5sqrt(3) + 2sqrt(3) = 7sqrt(6). Find the error.
Show Hint
When adding like radicals, what do you add?
Show Answer
Add coefficients, not radicands. Correct: 5sqrt(3) + 2sqrt(3) = 7sqrt(3)
Problem 4
Student work: (x + 4)(x - 4) = x2 + 16. Find the error.
Show Hint
This is the difference of squares pattern. What should the sign be?
Show Answer
Difference of squares: (a+b)(a-b) = a2 - b2. Correct: x2 - 16
Problem 5
Student work: sqrt(50) = sqrt(25) * sqrt(2) = 25sqrt(2). Find the error.
Show Hint
What is sqrt(25)?
Show Answer
sqrt(25) = 5, not 25. Correct: sqrt(50) = 5sqrt(2)
Problem 6
Student work: 3x2 * 2x3 = 6x6. Find the error.
Show Hint
When multiplying terms with the same base, what do you do with exponents?
Show Answer
Add exponents when multiplying: 2 + 3 = 5. Correct: 6x5
Problem 7
Student work: (4x3)2 = 8x6. Find the error.
Show Hint
What is 42?
Show Answer
42 = 16, not 8. Correct: 16x6
Problem 8
Student work: sqrt(x2 + 9) = x + 3. Find the error.
Show Hint
Test with x = 4: Is sqrt(16 + 9) = 4 + 3?
Show Answer
sqrt(a2 + b2) does not equal a + b. sqrt(x2 + 9) cannot be simplified further.
Problem 9
Student work: (5x + 3) - (2x - 7) = 3x - 4. Find the error.
Show Hint
Distribute the negative to both terms in the second polynomial.
Show Answer
-(2x - 7) = -2x + 7, not -2x - 7. Correct: 5x + 3 - 2x + 7 = 3x + 10
Problem 10
Student work: 2 / sqrt(2) = 2sqrt(2). Find the error.
Show Hint
Rationalize: multiply by sqrt(2)/sqrt(2). What's the denominator?
Show Answer
2/sqrt(2) * sqrt(2)/sqrt(2) = 2sqrt(2)/2 = sqrt(2). The student forgot to simplify.
Check Your Understanding
Answer these reflection questions about common mistakes.
1. What is the most common error when squaring binomials?
Show Answer
Forgetting the middle term (2ab) in the expansion.
2. Why is it important to study common mistakes before taking the SAT or ACT?
Show Answer
Wrong answer choices are often designed based on common student errors. Knowing these helps you avoid traps.
Next Steps
- Create a personal "error log" to track your own common mistakes
- Review these errors before any test or quiz
- Take the Unit Quiz to assess your mastery of all concepts