Polynomials Class 10 Mathematics — Revision Notes
This chapter covers polynomial expressions, their degrees, roots, and factorization methods. Students learn about zeros of polynomials, factor theorem, and
TL;DR: This chapter covers polynomial expressions, their degrees, roots, and factorization methods. Students learn about zeros of polynomials, factor theorem…
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This chapter covers polynomial expressions, their degrees, roots, and factorization methods. Students learn about zeros of polynomials, factor theorem, and
Polynomial Basics
- Polynomial is expression of form a0 + a1x + a2x^2 + ... + anxn where an is not zero
- Degree of polynomial is highest power of variable
- Linear polynomial: degree 1, Quadratic: degree 2, Cubic: degree 3
- Coefficient is numerical factor of each term
Zeros and Factor Theorem
- Zero of polynomial is value of variable making polynomial equal to zero
- Factor theorem: (x - a) is factor of p(x) if p(a) = 0
- Remainder theorem: when p(x) divided by (x - a), remainder is p(a)
- Quadratic p(x) = ax^2 + bx + c has two zeros (real or complex)
Factorization Methods
- Splitting middle term: for ax^2 + bx + c, find two numbers whose product is ac and sum is b
- Completing square: x^2 + bx + c = (x + b/2)^2 - b^2/4 + c
- Using identities: (a+b)^2, (a-b)^2, a^2-b^2, a^3+b^3, a^3-b^3
- Polynomial division: divide higher degree by lower degree polynomial
Relationship Between Zeros and Coefficients
- For quadratic ax^2 + bx + c with zeros r and s: r + s = -b/a, rs = c/a
- For cubic ax^3 + bx^2 + cx + d with zeros r, s, t: r+s+t = -b/a, rs+st+tr = c/a, rst = -d/a
- Sum of zeros = -b/a, Product of zeros = c/a for quadratic
- Can form polynomial if zeros are known
Key Terms
- Polynomial: Expression of form a0 + a1x + a2x^2 + ... with variables and coefficients
- Degree: Highest power of variable in polynomial
- Zero: Value of variable making polynomial equal to zero
Frequently Asked Questions
What is the relationship between zeros and coefficients?
For quadratic ax^2 + bx + c with zeros r and s: sum = -b/a and product = c/a. This helps find polynomial if zeros are known.
How do you factorize x^2 + 5x + 6?
Find two numbers whose product is 6 and sum is 5. These are 2 and 3. So x^2 + 5x + 6 = (x + 2)(x + 3).
More Class 10 Mathematics Revision Notes
- Real Numbers
- Pair of Linear Equations in Two Variables
- Quadratic Equations
- Arithmetic Progressions
- Triangles
- Coordinate Geometry
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