HW #31 - Intro to Polynomial Functions

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What is a polynomial function?

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A polynomial function is a mathematical expression that involves a sum of powers in one or more variables multiplied by coefficients. The general form is: f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

Front

What are the zeros of a polynomial?

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The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. They are also known as the roots of the polynomial.

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

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To find the factored form of a polynomial given its zeros, use the formula: f(x) = a(x - r1)(x - r2)...(x - rn), where r1, r2, ..., rn are the zeros.

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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. It indicates the maximum number of roots the polynomial can have.

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What is end behavior in polynomial functions?

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End behavior describes how the values of a polynomial function behave as x approaches positive or negative infinity. It is determined by the leading term of the polynomial.

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What does it mean if a polynomial has an even degree?

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If a polynomial has an even degree, its end behavior will be the same on both ends: as x approaches ±∞, f(x) will approach ±∞.

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What does it mean if a polynomial has an odd degree?

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If a polynomial has an odd degree, its end behavior will be opposite on both ends: as x approaches -∞, f(x) will approach -∞, and as x approaches +∞, f(x) will approach +∞.

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What is the standard form of a polynomial?

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The standard form of a polynomial is when it is written in descending order of the degree of its terms, starting with the highest degree term.

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How do you determine the maximum number of roots of a polynomial?

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The maximum number of roots of a polynomial is equal to its degree. For example, a polynomial of degree 3 can have up to 3 roots.

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What is the relationship between the degree of a polynomial and its number of turning points?

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The maximum number of turning points of a polynomial is one less than its degree. For example, a polynomial of degree 4 can have up to 3 turning points.