Solving Polynomial Equations

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What is a polynomial equation?

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A polynomial equation is an equation that involves a polynomial expression, which is a mathematical expression consisting of variables raised to whole number powers and coefficients.

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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 polynomial expression.

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How do you factor a polynomial?

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To factor a polynomial, you can look for common factors, use techniques like grouping, or apply special formulas such as the difference of squares or the sum/difference of cubes.

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution of a polynomial equation, in the form of a fraction p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.

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What is the significance of the Fundamental Theorem of Algebra?

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The Fundamental Theorem of Algebra states that every non-constant polynomial equation has at least one complex root, and thus a polynomial of degree n has exactly n roots (counting multiplicities).

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How do you solve a cubic equation?

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To solve a cubic equation, you can use factoring, synthetic division, or apply the cubic formula. Finding one root allows you to reduce the cubic to a quadratic.

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form (x - c), which is faster than long division.

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What are imaginary numbers?

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Imaginary numbers are numbers that can be expressed as a real number multiplied by the imaginary unit i, where i is defined as the square root of -1.

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What is the difference between real and complex roots?

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Real roots are solutions to polynomial equations that are real numbers, while complex roots include both real and imaginary parts, often occurring in conjugate pairs.

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What is the process to find the roots of a polynomial equation?

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To find the roots of a polynomial equation, you can set the polynomial equal to zero and solve for the variable using factoring, the quadratic formula, or numerical methods.