Graphing Polynomials Flashcard

Front

What is the general form of a polynomial equation?

Back

A polynomial equation is generally expressed as P(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 does it mean for a polynomial to be in factored form?

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A polynomial is in factored form when it is expressed as a product of its linear factors, such as y = a(x - r1)(x - r2)...(x - rn), where r1, r2, ..., rn are the roots or zeros of the polynomial.

Front

How can you determine the leading coefficient of a polynomial?

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The leading coefficient of a polynomial is the coefficient of the term with the highest degree (the term with the largest exponent).

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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 for which P(x) = 0. It represents the x-intercepts of the graph of the polynomial.

Front

How do you find the zeros of a polynomial given in factored form?

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To find the zeros of a polynomial in factored form, set each factor equal to zero and solve for x.

Front

What is the significance of the leading coefficient's sign in a polynomial?

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The sign of the leading coefficient determines the end behavior of the polynomial's graph: if positive, the graph rises to the right; if negative, it falls to the right.

Front

What is the degree of a polynomial, and how does it affect the graph?

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The degree of a polynomial is the highest exponent of x in the polynomial. It affects the number of zeros and the general shape of the graph.

Front

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

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A polynomial of degree n can have at most n real zeros, according to the Fundamental Theorem of Algebra.

Front

What is the relationship between the degree of a polynomial and its end behavior?

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The degree of a polynomial and the sign of the leading coefficient determine the end behavior: even degree with positive leading coefficient rises on both ends, odd degree with negative leading coefficient falls on the right.

Front

What is the process for sketching the graph of a polynomial function?

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To sketch the graph of a polynomial function, identify the zeros, determine the leading coefficient's sign, analyze the end behavior, and plot key points.