math 3 polynomial roots

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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. For example, 2x^2 + 3x + 1 is a polynomial.

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What are the roots of a polynomial?

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The roots (or zeros) of a polynomial are the values of x for which the polynomial equals zero. They can be real or complex numbers.

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How do you find the roots of a polynomial equation?

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To find the roots of a polynomial equation, you can use factoring, the quadratic formula, synthetic division, or numerical methods.

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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. For example, in 4x^3 + 2x^2 - x + 7, the degree is 3.

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What is the Fundamental Theorem of Algebra?

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The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n roots in the complex number system.

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What is multiplicity in relation to polynomial roots?

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Multiplicity refers to the number of times a particular root appears in a polynomial. For example, if (x - 2) is a factor of (x - 2)^3, then 2 is a root with multiplicity 3.

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How do you factor a difference of squares?

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A difference of squares can be factored using the formula a^2 - b^2 = (a - b)(a + b). For example, 49x^2 - 100 can be factored as (7x - 10)(7x + 10).

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor of the form (x - c).

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What is the relationship between the coefficients and the roots of a polynomial?

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The coefficients of a polynomial are related to its roots through Vieta's formulas, which relate the sums and products of the roots to the coefficients.

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How can you determine the number of real roots of a polynomial?

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The number of real roots can be determined using the discriminant (for quadratics) or by analyzing the polynomial's graph or using Descartes' Rule of Signs.