Unit 3 Extending Algebra Review

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or zero) of a polynomial equation, in the form of a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient.

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Define multiplicity in the context of polynomial functions.

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Multiplicity refers to the number of times a particular root (zero) appears in a polynomial. For example, if a polynomial has a factor of (x - r) raised to the power of n, then r is a root with multiplicity n.

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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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The possible Rational Zeros are: ±1, ±7, ±1/3, ±7/3.

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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 function equal to zero. It is also known as a root of the polynomial.

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How do you determine the multiplicity of a zero?

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To determine the multiplicity of a zero, factor the polynomial and observe the exponent of the factor corresponding to that zero. The exponent indicates the multiplicity.

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The degree of the polynomial is 7, which is the highest exponent of x.

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What does it mean for a polynomial to have complex solutions?

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Complex solutions are solutions that include imaginary numbers. They occur in conjugate pairs when the polynomial has real coefficients.

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If a polynomial has a zero at x = 2 with multiplicity 3, what does this imply about the graph of the polynomial at x = 2?

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The graph of the polynomial will touch the x-axis at x = 2 and turn around, indicating that it does not cross the x-axis at this point.