Precalc H Semester 1 Final Exam Review Units 1-3 (22-23)

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What are the types of discontinuities in a function?

Back

1. Jump Discontinuity: Occurs when the left-hand limit and right-hand limit exist but are not equal. 2. Infinite Discontinuity: Occurs when the function approaches infinity at a certain point. 3. Removable Discontinuity: Occurs when a limit exists at a point, but the function is not defined there.

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How do you find the zeros of a polynomial function?

Back

To find the zeros of a polynomial function, set the polynomial equal to zero and solve for the variable. This can involve factoring, using the quadratic formula, or synthetic division.

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What is a factor of a polynomial?

Back

A factor of a polynomial is a polynomial of lower degree that divides the polynomial without leaving a remainder. For example, if (x + 4) is a factor of f(x), then f(-4) = 0.

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What is the significance of the multiplicity of a root?

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The multiplicity of a root indicates how many times that root is repeated in the polynomial. A root with an even multiplicity will touch the x-axis and turn around, while a root with an odd multiplicity will cross the x-axis.

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How do you determine where a function is less than zero?

Back

To determine where a function is less than zero, find the intervals where the function's graph lies below the x-axis. This can be done by finding the zeros of the function and testing intervals between them.

Front

What is a polynomial function?

Back

A polynomial function is a function that can be expressed in the form f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

Front

What is the difference between a removable and non-removable discontinuity?

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A removable discontinuity can be 'fixed' by redefining the function at that point, while a non-removable discontinuity cannot be fixed and indicates a more severe break in the function.

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What is the Rational Root Theorem?

Back

The Rational Root Theorem states that any rational solution (or root) of a polynomial equation with integer coefficients is of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

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How do you use synthetic division to find polynomial zeros?

Back

To use synthetic division, set the polynomial equal to zero to find a potential root. Then, perform synthetic division with that root. If the remainder is zero, the root is confirmed.

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What is the Fundamental Theorem of Algebra?

Back

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.