Unit 1 Polynomial Functions 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: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer.

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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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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.

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How do you determine the number of real and complex roots of a polynomial?

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The number of real and complex roots can be determined using the degree of the polynomial and Descartes' Rule of Signs, which provides information about the possible number of positive and negative real roots.

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What is the factored form of a polynomial?

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The factored form of a polynomial expresses it as a product of its linear factors. For example, a polynomial \(f(x)\) can be expressed as \(f(x) = a(x - r_1)(x - r_2)...(x - r_n)\) where \(r_1, r_2, ..., r_n\) are the roots.

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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 function equal to zero. It corresponds to the x-intercepts of the graph of the function.

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How do you find the zeros of a polynomial?

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To find the zeros of a polynomial, set the polynomial equal to zero and solve for \(x\). This can involve factoring, using the quadratic formula, 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 expression. For example, in \(f(x) = 2x^3 + 3x^2 - x + 5\), the degree is 3.

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What is the significance of imaginary roots in polynomials?

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Imaginary roots occur in conjugate pairs for polynomials with real coefficients. If a polynomial has an imaginary root, it indicates that the polynomial does not cross the x-axis at that point.

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What is Descartes' Rule of Signs?

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Descartes' Rule of Signs is a method for determining the number of positive and negative real roots of a polynomial by counting the number of sign changes in the polynomial's coefficients.