Identify terms, factors and coefficients
Separate additive terms from multiplicative factors · Identify numerical coefficients · Evaluate an expression using its structure
Section Paper 3 · Historical Core · calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Basic algebraic notation
- Signed substitution
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Separate additive terms from multiplicative factors
- Identify numerical coefficients
- Evaluate an expression using its structure
Read the operations
Terms are separated by addition or subtraction, with each term's sign retained. In 3x²−5x+7, the terms are 3x², −5x and 7; numerical coefficients are 3 and −5 for the variable terms, with constant 7.
Factors multiply: in 4x(x+2), the factors include 4, x and (x+2). A bracket is one factor until expanded. A coefficient can depend on which variable is under discussion: in 3xy, the numerical coefficient is 3, while the coefficient of x with y fixed is 3y.
Substitution and equivalent structure
Use parentheses for negative substituted values and work through powers before products. Different equivalent forms can reveal different structures: 3x+6 and 3(x+2) have the same value, but the latter shows a common factor.
Sources for this lesson
Pause and explain
In 3x²−5x+7, what is the numerical coefficient of x?
Worked example
Identify the terms and coefficients in 5a²−2a+9.
Show the worked solution
- Separate the signed terms: 5a², −2a, 9.
- Variable-term numerical coefficients are 5 and −2; constant is 9.
Answer Terms 5a², −2a, 9; coefficients 5 and −2; constant 9.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Identify the signed terms and constant in 2y²+4y−6.
Give me a hint
Separate at addition/subtraction while retaining signs.
Compare my reasoning
- Terms are 2y², 4y, −6.
- The constant is −6.
2y², 4y, −6; constant −6.
Look for these in your work
- Retain signs
- Distinguish a constant from a variable term
List the displayed factors in 6a(a−1), then evaluate at a = 2.
Give me a hint
A bracket is a factor.
Compare my reasoning
- Factors are 6, a, (a−1).
- At 2, 6×2×(2−1) = 12.
Factors 6, a, (a−1); value 12.
Look for these in your work
- Read multiplication
- Substitute into every occurrence
A fee is 4n+12 for n items. Explain each term and evaluate for n = 5.
Give me a hint
Separate variable and fixed charges.
Compare my reasoning
- 4n is 4 per item; 12 is a fixed fee.
- At n = 5 the total is 20+12 = 32.
4 per item plus fixed 12; total 32 for 5 items.
Look for these in your work
- Interpret terms in context
- Check the full total
Common mistakes
- The minus sign belongs to a negative term.
- Terms add; factors multiply.
What can you explain now?
In 3x²−5x+7, what is the numerical coefficient of x?
Compare with the explanation
−5.
The subtraction sign is part of the x term.
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Make it stick
In 3x²−5x+7, what is the numerical coefficient of x?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.