ACC Geo Unit 5 Part 2 - Flashcard Review

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What is a polynomial function?

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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is P(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

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What does it mean for a polynomial to have integral coefficients?

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A polynomial has integral coefficients if all the coefficients (the numbers in front of the variables) are integers.

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How do you determine 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 P(x) = 2x^3 - 4x^2 + 5, the degree is 3.

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What is a zero of a polynomial function?

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A zero of a polynomial function is a value of x that makes the polynomial equal to zero. It is also known as a root of the polynomial.

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How many zeros can a polynomial of degree n have?

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A polynomial of degree n can have at most n zeros, counting multiplicities.

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What is the significance of complex zeros in polynomials?

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Complex zeros occur in conjugate pairs. If a polynomial has real coefficients, any non-real zeros must appear as pairs of complex conjugates.

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

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If r is a zero of a polynomial P(x), then (x - r) is a factor of P(x). This means that the polynomial can be expressed as a product of its factors.

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How do you find a polynomial function given its zeros?

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To find a polynomial function given its zeros, write the polynomial as a product of factors corresponding to each zero. For example, if the zeros are r1, r2, and r3, then P(x) = k(x - r1)(x - r2)(x - r3), where k is a constant.

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What is the multiplicity of a zero?

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The multiplicity of a zero refers to the number of times that zero appears as a root of the polynomial. A zero with an odd multiplicity will cross the x-axis, while a zero with an even multiplicity will touch the x-axis.

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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 function of degree n has exactly n roots in the complex number system, counting multiplicities.