Polynomials

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A polynomial in one variable x is an algebraic expression of the form anxn + an-1xn-1 + ... + a2x2 + a1x + a0, where a0, a1, a2, ..., an are constants and an ≠ 0. All exponents must be whole numbers.

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Examples: x + 1/x, √x + 3, 3y + y². These are not polynomials because: (1) x + 1/x = x + x⁻¹ has negative exponent, (2) √x + 3 = x^(1/2) + 3 has fractional exponent, (3) 3y + y² has irrational exponen…

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Monomial: Polynomial with one term (e.g., 2x, 5x³, -7) Binomial: Polynomial with two terms (e.g., x + 1, x² - x, y⁹ + 1) Trinomial: Polynomial with three terms (e.g., x² + x + π, 2 + x - x², y⁴ + y + …

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The degree of a polynomial is the highest power of the variable in the polynomial. For 3x⁷ - 4x⁶ + x + 9, the term with highest power is 3x⁷, so the degree is 7. Note: The degree of a non-zero constan…

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A linear polynomial is a polynomial of degree one. General form: ax + b, where a ≠ 0 and a, b are constants. Examples: 4x + 5, 2y, t + 2, 3 - u. Note: A linear polynomial can have at most two terms.

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A quadratic polynomial is a polynomial of degree two. General form: ax² + bx + c, where a ≠ 0 and a, b, c are constants. Examples: 2x² + 5, 5x² + 3x + π, x², 5 - y², 4y + 5y². Note: A quadratic polyno…

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A cubic polynomial is a polynomial of degree three. General form: ax³ + bx² + cx + d, where a ≠ 0 and a, b, c, d are constants. Examples: 4x³, 2x³ + 1, 5x³ + x², 6x³ - x, 6 - x³, 2x³ + 4x² + 6x + 7. N…

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A zero of polynomial p(x) is a number c such that p(c) = 0. To find zeros: Set p(x) = 0 and solve for x. Example: For p(x) = x - 1, setting x - 1 = 0 gives x = 1, so 1 is the zero. The zero of a polyn…