Questions
3–5 questions per board paper
Difficulty
Medium
Importance
Core — never skip
Overview
Polynomials represent algebraic expressions involving variables and coefficients, forming the foundation of algebra in the CBSE curriculum. Mastering zeroes and their relationship with coefficients is crucial, as these concepts frequently appear in high-weightage sections of board exams. Students must focus on the structural behavior of polynomials to solve complex problems efficiently.
Fundamentals of Polynomials
A polynomial is an expression where exponents of the variable are non-negative integers. Understanding the degree of a polynomial is essential for identifying the maximum number of zeroes possible.
- Standard form: p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0
- Degree: Highest power of the variable x
- Linear polynomial: degree 1
- Quadratic polynomial: degree 2
- Cubic polynomial: degree 3
Zeroes of a Polynomial
The zero of a polynomial is the value of the variable that makes the entire expression evaluate to zero. Geometrically, this corresponds to the x-coordinate of the point where the polynomial graph intersects the x-axis.
- Zero of linear p(x) = ax + b is x = -b/a
- A polynomial of degree n has at most n zeroes
- Number of zeroes equals number of x-axis intersections
- For p(x) = k, graph is parallel to x-axis
- Verification: p(zero) must equal 0
Relationship between Zeroes and Coefficients
For a quadratic polynomial, the relationship between zeroes (alpha and beta) and coefficients (a, b, c) is a standard examination requirement. These identities are frequently tested in objective and short-answer formats.
- Sum of zeroes: alpha + beta = -b/a
- Product of zeroes: alpha * beta = c/a
- Quadratic form: x^2 - (sum of zeroes)x + (product of zeroes)
- For cubic: alpha + beta + gamma = -b/a
- For cubic: alpha*beta + beta*gamma + gamma*alpha = c/a
- For cubic: alpha*beta*gamma = -d/a
Formula Sheet
p(x) = a_nx^n + ... + a_0
x = -b/a (Linear)
Sum of zeroes = -b/a
Product of zeroes = c/a
p(x) = k[x^2 - (sum)x + (product)]
alpha + beta + gamma = -b/a
alpha*beta + beta*gamma + gamma*alpha = c/a
alpha*beta*gamma = -d/a
Exam Tip
Always rearrange the polynomial into standard descending order of powers before extracting coefficients (a, b, c) to avoid sign errors.
Common Mistakes
- Swapping the negative sign for sum of zeroes: -b/a is often miscalculated as b/a.
- Forgetting to verify the degree before assuming the number of zeroes.
- Misidentifying coefficients in a non-standard polynomial equation (e.g., failing to reorder terms by power).
More Revision Notes
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