Write polynomial functions given its roots

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

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What are the roots of a polynomial function?

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The roots (or zeros) of a polynomial function are the values of x for which the polynomial equals zero. They represent the x-intercepts of the graph of the polynomial.

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How do you find a polynomial function given its roots?

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To find a polynomial function given its roots, use the fact that if r1, r2, ..., rn are the roots, the polynomial can be expressed as f(x) = a(x - r1)(x - r2)...(x - rn), where a is a non-zero constant.

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What is the least degree of a polynomial function?

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The least degree of a polynomial function is the smallest degree of the polynomial that can be formed using the given roots. It is equal to the number of distinct roots.

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Write a polynomial function of least degree with roots 4, -4, i, -i.

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f(x) = x^4 - 15x^2 - 16.

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Write a polynomial function of least degree with roots 2, 1, -1.

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f(x) = x^3 - 2x^2 - x + 2.

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Write a polynomial function of least degree with roots 4, -1, 6.

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f(x) = x^3 - 9x^2 + 14x + 24.

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Write a polynomial function of least degree with roots 3, -1, 1, 2.

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f(x) = x^4 - 5x^3 + 5x^2 + 5x - 6.

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Write a polynomial function of least degree with roots -2, 5, -3i, 3i.

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f(x) = x^4 - 3x^3 - x^2 - 27x - 90.

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What is the significance of integral coefficients in polynomial functions?

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Integral coefficients in polynomial functions mean that all the coefficients of the polynomial are integers. This is important for ensuring that the polynomial can be expressed in a standard form with whole number coefficients.