2023 PC Semester 1 REVIEW

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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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Define positive, negative, and imaginary zeros of a function.

Back

Positive zeros are the roots of the function that are greater than zero, negative zeros are the roots that are less than zero, and imaginary zeros are roots that have no real part (e.g., they are of the form a + bi where b ≠ 0).

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How do you determine the intervals of increase and decrease for a function?

Back

To determine the intervals of increase and decrease, find the derivative of the function. If the derivative is positive in an interval, the function is increasing; if negative, the function is decreasing.

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What is a decreasing interval in a function?

Back

A decreasing interval is a range of x-values where the function's output (y-value) decreases as x increases.

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What is the significance of the degree of a polynomial?

Back

The degree of a polynomial indicates the highest power of the variable in the polynomial, which determines the polynomial's end behavior and the maximum number of roots it can have.

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Explain the concept of multiplicity of roots.

Back

Multiplicity of roots refers to the number of times a particular root appears in a polynomial. A root with an even multiplicity will touch the x-axis, while a root with an odd multiplicity will cross the x-axis.

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What is the relationship between the coefficients of a polynomial and its roots?

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The coefficients of a polynomial are related to its roots through Vieta's formulas, which express the sums and products of the roots in terms of the coefficients.

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Define what a polynomial function is.

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What is the purpose of factoring polynomials?

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Factoring polynomials is used to simplify expressions, solve polynomial equations, and find the roots of the polynomial.