Unit 5 Test Review Polynomial Functions_Jenkins SY25

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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_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

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What are the roots of a polynomial function?

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The roots (or zeros) of a polynomial function are the values of x for which the function equals zero. They can be real or complex numbers.

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What does multiplicity mean in the context of polynomial roots?

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Multiplicity refers to the number of times a particular root appears in a polynomial. For example, if (x - r) appears twice in the factorization, r is a root with multiplicity 2.

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How do you find the roots of a polynomial function?

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To find the roots, set the polynomial equal to zero and solve for x. This can involve factoring, using the quadratic formula, or applying synthetic division.

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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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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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What is a complex root?

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A complex root is a solution to a polynomial equation that includes imaginary numbers. Complex roots occur in conjugate pairs when coefficients are real.

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What is the significance of the leading coefficient in a polynomial function?

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The leading coefficient determines the end behavior of the polynomial graph. If it is positive, the graph rises to the right; if negative, it falls to the right.

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How do you determine the number of real solutions from a polynomial's graph?

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The number of times the graph intersects the x-axis indicates the number of real solutions. Each intersection corresponds to a real root.

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What is the conjugate root theorem?

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The conjugate root theorem states that if a polynomial has real coefficients and a complex root a + bi, then its conjugate a - bi is also a root.