5.2 Writing Polynomial Equations Given Roots

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What is a polynomial equation?

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A polynomial equation is an equation that involves a polynomial expression, which is a sum of terms, each consisting of a variable raised to a non-negative integer power multiplied by a coefficient.

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What are the roots of a polynomial?

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The roots of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as zeros of the polynomial.

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How do you write a polynomial in factored form given its roots?

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To write a polynomial in factored form given its roots, use the formula: (x - r1)(x - r2)...(x - rn) where r1, r2, ..., rn are the roots.

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What is the significance of multiplicity in polynomial roots?

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Multiplicity refers to the number of times a particular root appears in a polynomial. A root with multiplicity greater than 1 indicates that the polynomial touches the x-axis at that root.

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If a polynomial has a root of 5, what does this imply about the polynomial?

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If a polynomial has a root of 5, it means that (x - 5) is a factor of the polynomial.

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What is the relationship between complex roots and their conjugates in polynomials?

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In polynomials with real coefficients, complex roots occur in conjugate pairs. If a polynomial has a complex root a + bi, it must also have the root a - bi.

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How do you determine the degree of a polynomial from its factored form?

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The degree of a polynomial is the highest power of the variable in the polynomial. In factored form, it is the sum of the multiplicities of all the roots.

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What is the simplest polynomial function with roots 7 (multiplicity 2) and -3?

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How do you find the roots of a polynomial given in factored form?

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To find the roots of a polynomial in factored form, set each factor equal to zero and solve for the variable.

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What does it mean for a root to have a multiplicity of 3?

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A root with a multiplicity of 3 means that the root appears three times in the polynomial, indicating that the graph of the polynomial will touch the x-axis at that root and turn around.