2024 PreCal Semester 1 EXAM

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form (x - c). It is faster than long division and is used primarily for polynomials.

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What is the average rate of change of a function?

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The average rate of change of a function f(x) over an interval [a, b] is given by the formula: \( \frac{f(b) - f(a)}{b - a} \). It represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

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What does end behavior of a function refer to?

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End behavior describes how a function behaves as the input values approach positive or negative infinity. It helps in determining the limits of the function at the extremes.

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How do you factor the difference of squares?

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The difference of squares can be factored using the formula: \( a^2 - b^2 = (a + b)(a - b) \). For example, \( 4x^2 - 25 = (2x + 5)(2x - 5) \).

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What is an absolute minimum in a graph?

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An absolute minimum is the lowest point on the graph of a function over its entire domain. It is the smallest value that the function attains.

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What is the significance of the leading coefficient in polynomial functions?

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

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What is a point of inflection?

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A point of inflection is a point on the graph of a function where the concavity changes. It is where the second derivative is zero or undefined.

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How do you find the roots of a polynomial?

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

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What is the difference between local and absolute extrema?

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Local extrema are the highest or lowest points in a specific interval, while absolute extrema are the highest or lowest points over the entire domain of the function.

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What is the purpose of the Remainder Theorem?

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The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is f(c). This helps in evaluating polynomials at specific points.