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Knowledge
Polynomials are essential mathematical expressions characterized by their degrees. This chapter explores the definitions, types, and geometric interpretations of polynomials, particularly focusing on their zeroes and the relationship between zeroes and coefficients. It also discusses the division algorithm for polynomials and highlights key concepts through various examples.
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Sections
References
NCERT Study MaterialClass Notes
Memorization
What we have learnt
- Polynomials of degrees 1, 2...
- A quadratic polynomial in x...
- The zeroes of a polynomial ...
Final Test
Revision Tests
Chapter FAQs
What we have learnt
- Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
- A quadratic polynomial in x with real coefficients is of the form ax2 + bx + c, where a, b, and c are real numbers with a ≠ 0.
- The zeroes of a polynomial p(x) correspond to the x-coordinates of points where the graph of y = p(x) intersects the x-axis.
- A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.
- If α and β are the zeroes of a quadratic polynomial ax2 + bx + c, then the relationships are given by -b/a = α + β and c/a = αβ.
- For cubic polynomials, there are established relationships among the coefficients and zeroes: -b/a = α + β + γ, c/a = αβ + βγ + γα, and -d/a = αβγ.
Key Concepts
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Term: Polynomial
Definition: An expression consisting of variables raised to whole number powers, combined using addition, subtraction, multiplication, and coefficients.
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Term: Linear Polynomial
Definition: A polynomial of degree 1 that has the general form ax + b.
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Term: Quadratic Polynomial
Definition: A polynomial of degree 2 expressed in the form ax2 + bx + c where a ≠ 0.
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Term: Cubic Polynomial
Definition: A polynomial of degree 3 expressed in the form ax3 + bx2 + cx + d where a ≠ 0.
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Term: Zeroes of a Polynomial
Definition: The values of x that make the polynomial equal to zero, represented by the x-coordinates where the graph intersects the x-axis.