Writing Polynomial Functions

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What is a polynomial function?

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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_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where n is a non-negative integer.

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What does it mean for a polynomial to be in standard form?

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A polynomial is in standard form when it is written as a sum of terms in descending order of the degree of each term, starting from the highest degree to the constant term.

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How do you find the zeros of a polynomial function?

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The zeros of a polynomial function are the values of x for which f(x) = 0. They can be found by factoring the polynomial or using the quadratic formula for polynomials of degree 2.

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

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Each zero of a polynomial corresponds to a factor of the form (x - zero). For example, if x = 3 is a zero, then (x - 3) is a factor.

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How do you write a polynomial given its zeros?

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To write a polynomial given its zeros, use the factored form: f(x) = a(x - r1)(x - r2)...(x - rn), where r1, r2, ..., rn are the zeros and a is a non-zero constant.

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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 f(x) = 2x^3 + 3x^2 - x + 5, the degree is 3.

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What is a leading coefficient?

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The leading coefficient is the coefficient of the term with the highest degree in a polynomial. For example, in f(x) = 4x^3 + 2x^2 - 5, the leading coefficient is 4.

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How do you convert a polynomial from factored form to standard form?

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To convert from factored form to standard form, expand the factors using the distributive property (FOIL method for binomials) and combine like terms.

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What is the factored form of a polynomial?

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The factored form of a polynomial expresses it as a product of its factors. For example, f(x) = (x - 1)(x + 2) is a factored form.

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What is the significance of the roots of a polynomial?

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The roots (or zeros) of a polynomial indicate the x-values where the graph of the polynomial intersects the x-axis.