Alg2 Polynomial Roots & Zeros

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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 of a polynomial are the values of $$x$$ for which the polynomial equals zero. They are also known as zeros of the polynomial.

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

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The degree of a polynomial indicates the maximum number of roots it can have. For example, a polynomial of degree 3 can have up to 3 roots.

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Define multiplicity in the context of polynomial roots.

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Multiplicity refers to the number of times a particular root appears in a polynomial. A root with a multiplicity greater than 1 is called a repeated root.

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How do you find a polynomial given its roots?

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To find a polynomial given its roots, use the fact that if $$r_1, r_2, ..., r_n$$ are the roots, the polynomial can be expressed as: $$f(x) = a(x - r_1)(x - r_2)...(x - r_n)$$ where $$a$$ is a non-zero constant.

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What is the significance of complex roots in polynomials?

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If a polynomial has real coefficients, then any complex roots must occur in conjugate pairs. For example, if $$a + bi$$ is a root, then $$a - bi$$ is also a root.

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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 determine the number of real roots of a polynomial?

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The number of real roots can be determined using the Descartes' Rule of Signs, which analyzes the number of sign changes in the polynomial's coefficients.

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What is a polynomial of least degree?

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A polynomial of least degree is the simplest polynomial that has the given roots, meaning it has the smallest degree possible.