A cubic polynomial is a polynomial of degree three expressed in the form ax³ + bx² + cx + d where a b c and d are constants and a ≠ 0. It has at most three real roots and its graph is a curve with possible turning points. Solving cubic polynomials involves factorization synthetic division or formulas. They are vital in algebra calculus and engineering applications.
Class 10 Maths Chapter 2 Polynomials covers key concepts like zeros of polynomials relationships between coefficients and zeros and division algorithm for polynomials. It focuses on linear quadratic and cubic polynomials. This chapter strengthens algebraic skills essential for CBSE Exam 2024-25 and helps in solving real-life problems through polynomial equations ensuring a deeper understanding of mathematical relationships and applications.
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Understanding Zeros of a Cubic Polynomial
Mathematical Background
A cubic polynomial is of the general form:
ax³ + bx² + cx + d, where a ≠ 0
Fundamental Theorem of Algebra
– Any polynomial has exactly as many zeros as its degree
– These zeros can be real or complex numbers
– In a cubic polynomial, these zeros are referred to as “roots”
Mere Zero Analysis
– A cubic polynomial ALWAYS has 3 zeros
– These zeros can include:
– 3 real zeros in different positions
– 1 real zero and 2 complex conjugate zeros
– 1 repeated real zero that occurs twice
– One real zero that occurs thrice
Mathematical Proof Highlights
– The Fundamental Theorem of Algebra assures 3 zeros
– Complex numbers make all polynomials fully factorable
– Mathematically denoted as:
ax³ + bx² + cx + d = a(x – r₁)(x – r₂)(x – r₃)
Where r₁, r₂, r₃ are the three zeros
Key Insight
The degree of the polynomial always decides the number of zeros,
never mind the actual type of those zeros.
Conclusion:
A cubic polynomial ALWAYS has 3 zeros.
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