Polynomials

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A polynomial is an algebraic expression consisting of variables and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents. Example: 2x³ - 5x² + 3x

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The degree of a polynomial is the highest power of the variable in the polynomial. For 5x³ - 4x² + x - √2, the highest power of x is 3, so the degree is 3.

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A linear polynomial is a polynomial of degree 1. Examples: (1) 2x - 3, (2) √3x + 5, (3) y + √2. General form: ax + b where a ≠ 0.

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A quadratic polynomial is a polynomial of degree 2. General form: ax² + bx + c where a ≠ 0. Examples: (1) 2x² + 3x - 2/5, (2) y² - 2. The name comes from 'quadrate' meaning square.

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A cubic polynomial is a polynomial of degree 3. General form: ax³ + bx² + cx + d where a ≠ 0. Examples: 2 - x³, x³, √2x³, 3x³ - 2x² + x - 1.

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The value of polynomial p(x) at x = k is denoted as p(k) and obtained by substituting x = k in p(x). For p(x) = x² - 3x - 4: p(2) = (2)² - 3(2) - 4 = 4 - 6 - 4 = -6.

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A real number k is called a zero of polynomial p(x) if p(k) = 0. Example: For p(x) = x² - 3x - 4, we have p(-1) = (-1)² - 3(-1) - 4 = 1 + 3 - 4 = 0, so -1 is a zero of this polynomial.

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For linear polynomial ax + b (where a ≠ 0), if k is a zero, then ak + b = 0, which gives k = -b/a. Therefore, the zero of linear polynomial ax + b is -b/a = -(constant term)/(coefficient of x).