Identities and coefficients
Objective. Use identities to compare coefficients and determine unknown constants.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
An identity is a statement that is true for every value of the variable, shown using the symbol ≡, and this topic uses identities to find unknown coefficients by comparing matching terms on each side. If two expressions are identically equal, the coefficients of each power of x must match exactly, such as x² terms matching x² terms and constants matching constants. For example, if 2x² + ax + 3 ≡ 2x² − 6x + b, comparing x terms gives a = −6, and comparing constants gives b = 3. Expanding one side fully before comparing is essential for this method to work.
Worked example
Given that 3x² + bx − 5 ≡ 3x² − 9x + c for all x, find the values of the identity's constants b and c.
- 1Compare the x-coefficients on both sides: b = −9.
- 2Compare the constant terms on both sides: −5 = c.
- 3So c = −5.
- 4Final values: b = −9 and c = −5.
Answer: b = −9, c = −5
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Pupils substitute a single number into an identity to try to find two unknowns at once, which only gives one equation. Always compare coefficients of matching powers separately to get enough equations.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.