Polynomials — Class 10 CBSE Mathematics revision notes
A 4-page visual revision sheet for Polynomials, written for Class 10 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Polynomials in one look
the zeroes and the coefficients are two views of one thing
What counts as a polynomial
- an expression p(x)=a_nx^n+…+a_1x+a_0 in which every power of x is a whole number
- the degree is the highest power of x that actually shows up
- degree 1 is linear, degree 2 is quadratic, degree 3 is cubic
- √x and 1/x are shut out: their powers are 1/2 and -1, not whole numbers
Value at a point, and a zero
- the value of p at x=k
p(k) - k is a zero of p exactly when
p(k)=0 - zero of the linear ax+b
x=-b/a - check on 2x-6
p(3)=2(3)-6=0
Named by degree
| degree | general form | zeroes |
|---|---|---|
| 0, constant | p(x)=5 | none at all |
| 1, linear | ax+b, a≠0 | exactly one |
| 2, quadratic | ax^2+bx+c | at most two |
| 3, cubic | ax^3+bx^2+cx+d | at most three |
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Polynomials sheet covers
- Introduction
- What counts as a polynomial
- Value at a point, and a zero
- the value of p at x=k
- k is a zero of p exactly when
- zero of the linear ax+b
- check on 2x-6
- Named by degree
- Geometrical Meaning of the Zeroes of a Polynomial
- Geometrical meaning of the zeroes
- One zero, and none at all
- When a sketch is all you are given
- Relationship between Zeroes and Coefficients of a Polynomial
- Zeroes and coefficients of a quadratic
- for ax^2+bx+c with zeroes α and β
- sum of the zeroes
- product of the zeroes
- check on x^2-5x+6, zeroes 2 and 3
- Where those two relations come from
- a quadratic with zeroes α and β
- multiply the brackets out
- line it up against ax^2+bx+c
- divide by a and there they are
- Building a quadratic from its zeroes
- from sum S and product P
- zeroes 2 and 3
- zeroes -3 and 5
- k=2 gives the same zeroes
- Splitting the middle term
- What the two signs alone tell you
- Expressions rebuilt from sum and product
- squares
- difference, up to sign
- reciprocals
- on x^2-5x+6, zeroes 2 and 3
- One quadratic, worked end to end
- 2x²-x-6, on the axis
- Three shapes worth recognising on sight
- a difference of two squares
- so its zeroes sit either side of 0
- a perfect square, one repeated zero
- and a plain common factor
- When a sentence hides a coefficient
- Symbols
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Polynomials the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.