Polynomial Division Review

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, f(x) = 2x^3 - 4x^2 + 3 is a polynomial.

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What does it mean for a polynomial to have a factor?

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A polynomial f(x) has a factor g(x) if there exists another polynomial q(x) such that f(x) = g(x) * q(x). This means that when f(x) is divided by g(x), the remainder is zero.

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What is the Factor Theorem?

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The Factor Theorem states that if f(c) = 0 for a polynomial f(x), then (x - c) is a factor of f(x).

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How do you determine if (x - c) is a factor of a polynomial?

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To determine if (x - c) is a factor of a polynomial f(x), evaluate f(c). If f(c) = 0, then (x - c) is a factor.

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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 factor of the form (x - c). It is faster than long division and is used when the divisor is a linear polynomial.

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What is the Remainder Theorem?

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The Remainder Theorem states that the remainder of the division of a polynomial f(x) by (x - c) is equal to f(c).

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How do you find the remainder when dividing a polynomial by a linear factor?

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To find the remainder when dividing a polynomial f(x) by (x - c), simply evaluate f(c).

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What is the significance of the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial. It determines the polynomial's behavior and the number of roots it can have.

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What is a zero of a polynomial?

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A zero of a polynomial f(x) is a value c such that f(c) = 0. Zeros are also the x-intercepts of the polynomial's graph.