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, f(x) = 2x^3 - 4x + 7 is a polynomial.

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What are the roots (or zeros) 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 are the points where the graph of the polynomial intersects the x-axis.

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How do you determine the number of zeros of a polynomial function?

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The number of zeros of a polynomial function is determined by its degree. A polynomial of degree n can have up to n real roots.

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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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What is the Remainder Theorem?

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The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is f(c).

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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 = (7x - 10)(7x + 10).

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What is the standard form of a polynomial?

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The standard form of a polynomial is expressed as a sum of terms in descending order of the degree of the variable. For example, f(x) = 3x^4 + 2x^3 - x + 5.

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How do you find the zeros of a polynomial function?

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To find the zeros of a polynomial function, set the polynomial equal to zero and solve for x. This may involve factoring, using the quadratic formula, or other methods.

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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 to find polynomial quotients and remainders.

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What is the relationship between the degree of a polynomial and its number of zeros?

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The degree of a polynomial indicates the maximum number of zeros it can have. A polynomial of degree n can have up to n real zeros, counting multiplicities.