2. POLYNOMIALS
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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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
- -- Polynomial
- An expression consisting of variables raised to whole number powers, combined using addition, subtraction, multiplication, and coefficients.
- -- Linear Polynomial
- A polynomial of degree 1 that has the general form ax + b.
- -- Quadratic Polynomial
- A polynomial of degree 2 expressed in the form ax2 + bx + c where a ≠ 0.
- -- Cubic Polynomial
- A polynomial of degree 3 expressed in the form ax3 + bx2 + cx + d where a ≠ 0.
- -- Zeroes of a Polynomial
- The values of x that make the polynomial equal to zero, represented by the x-coordinates where the graph intersects the x-axis.
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