Polynomials

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A polynomial in x is an algebraic expression containing the sum of a finite number of terms of the form ax^n, where 'a' is a real number, a ≠ 0, and 'n' is a whole number. Example: 3x² + 5x - 2

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Linear polynomial: degree 1 (e.g., 3x + 5) Quadratic polynomial: degree 2 (e.g., x² + 5x + 4) Cubic polynomial: degree 3 (e.g., 2x³ - 4x² + x - 1)

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The degree of a polynomial p(x) is the highest power of x in p(x). For example, in 5x³ - 4x² + x - 1, the degree is 3.

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A real number k is said to be a zero of polynomial p(x) if p(k) = 0. For example, if p(x) = x² - 5x + 6, then p(2) = 0 and p(3) = 0, so 2 and 3 are zeros of the polynomial.

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The zeros of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the X-axis. These are the points where the polynomial value becomes zero.

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A linear polynomial ax + b (a ≠ 0) has exactly one zero, which is x = -b/a. Geometrically, the graph y = ax + b intersects the X-axis at exactly one point.

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A quadratic polynomial can have: • Two distinct zeros (parabola cuts X-axis at two points) • One repeated zero (parabola touches X-axis at one point) • No real zeros (parabola doesn't intersect X-axis

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If α and β are zeros of ax² + bx + c (a ≠ 0), then: Sum of zeros: α + β = -b/a Product of zeros: αβ = c/a