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Find all the zeros. Write the answer in exact form. \[ n(x)=x^{3}-7 x^{2}+11 x+3 \] If there is more than one answer, separate them with commas. Select "None" if applicable. The zeros of \( n(x): \square \)

Ask by Lambert Welch. in the United States
Mar 15,2025

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The zeros of \( n(x) \) are \( 2 - \sqrt{5} \), \( 3 \), and \( 2 + \sqrt{5} \).

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To find the zeros of the polynomial \( n(x) = x^{3}-7 x^{2}+11 x+3 \), you can use the Rational Root Theorem to test possible rational roots that could be factors of the constant term (3) over the leading coefficient (1). After evaluating potential rational roots, you might discover that \( x = 1 \) is a root. Using synthetic division, divide \( n(x) \) by \( x-1 \) to find the quotient \( x^{2}-6 x-3 \). You can then use the quadratic formula on \( x^{2}-6 x-3 \) to find the remaining roots. The final zeros are: \[ x = 1, 3 + 2\sqrt{2}, 3 - 2\sqrt{2} \]

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