Chapter 2 Precalculus Assessment Part 1

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What is a polynomial function?

Back

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $$a_n, a_{n-1}, ..., a_0$$ are constants and $$n$$ is a non-negative integer.

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in $$f(x) = 3x^4 + 2x^3 - x + 5$$, the degree is 4.

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What is a zero of a function?

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A zero of a function is a value of $$x$$ for which the function evaluates to zero, i.e., $$f(x) = 0$$. It represents the x-intercepts of the graph of the function.

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How do you find the y-intercept of a function?

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To find the y-intercept of a function, evaluate the function at $$x = 0$$. The y-intercept is the point where the graph crosses the y-axis.

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What does it mean for a zero to 'bounce' off the x-axis?

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A zero 'bounces' off the x-axis if it has an even multiplicity. This means that the graph touches the x-axis at that point but does not cross it.

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What is the x-intercept of a function?

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The x-intercept of a function is the point(s) where the graph of the function crosses the x-axis, which corresponds to the zeros of the function.

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

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The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It determines the end behavior of the graph of the polynomial.

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What is the difference between a root and a zero?

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In the context of functions, 'root' and 'zero' are often used interchangeably to refer to the values of $$x$$ that make the function equal to zero.

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How do you determine the number of zeros a polynomial can have?

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The Fundamental Theorem of Algebra states that a polynomial of degree $$n$$ can have at most $$n$$ real zeros.

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What is the process for factoring a polynomial?

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Factoring a polynomial involves expressing it as a product of its factors. This can be done using methods such as grouping, synthetic division, or applying the quadratic formula.