Writing 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, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

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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 roots or solutions of the polynomial equation.

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How do you find the polynomial from its zeros?

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To find a polynomial from its zeros, use the fact that if r is a zero, then (x - r) is a factor of the polynomial. Multiply the factors corresponding to each zero to obtain the polynomial.

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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 expressed as a sum of terms in descending order of the degree of each term, typically written as a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0.

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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. For example, in the polynomial 3x^4 + 2x^2 - 5, the degree is 4.

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What does it mean for a polynomial to have integral coefficients?

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A polynomial has integral coefficients if all the coefficients (the numbers in front of the variable terms) are integers. For example, 2x^3 - 3x + 5 has integral coefficients.

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How do you write a polynomial function with given zeros?

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To write a polynomial function with given zeros, create factors for each zero (x - zero) and multiply them together. For example, for zeros 2 and -3, the polynomial is f(x) = (x - 2)(x + 3).

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What is a multiplicity of a zero?

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The multiplicity of a zero refers to the number of times a particular zero appears as a factor of the polynomial. For example, if (x - 1)^2 is a factor, then 1 has a multiplicity of 2.

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

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A polynomial of degree n can have at most n zeros, counting multiplicities. For example, a cubic polynomial (degree 3) can have up to 3 zeros.

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What is the relationship between the zeros and the factors of a polynomial?

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The zeros of a polynomial correspond to the values of x that make the polynomial equal to zero, and each zero (r) corresponds to a factor of the form (x - r).