Solving polynomials

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (or indeterminates) raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial 3x^3 - 4x^2 + 2, the degree is 3.

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What is a zero of a polynomial?

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A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. For example, if f(x) = x^2 - 4, then x = 2 and x = -2 are zeros.

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or root) of a polynomial equation with integer coefficients is of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

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How do you factor a quadratic polynomial?

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To factor a quadratic polynomial of the form ax^2 + bx + c, find two numbers that multiply to ac and add to b. Rewrite the polynomial using these numbers and factor by grouping.

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor of the form (x - c). It is faster and requires less writing.

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What is the relationship between the zeros and the factors of a polynomial?

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The zeros of a polynomial correspond to the values of x that make the polynomial equal to zero, and each zero (r) can be expressed as a factor of the polynomial in the form (x - r).

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What is the significance of the leading coefficient in a polynomial?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It affects the end behavior of the polynomial's graph.

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What is the difference between rational and irrational roots?

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Rational roots can be expressed as a fraction of two integers, while irrational roots cannot be expressed as a simple fraction and often involve square roots or other non-repeating decimals.

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How can you determine if a polynomial has complex roots?

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A polynomial has complex roots if its discriminant (b^2 - 4ac for a quadratic) is negative, indicating that the polynomial does not intersect the x-axis.