Pregunta
5. Find a polynomial \( f(x) \) with a leading coefficient of 2 and a zero at -3 with multiplicity 2 and a zero at 4 with multiplicity 3 . Write your answer in complete factored form.
Ask by Christensen Allan. in Germany
Mar 13,2025
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The polynomial is \( f(x) = 2(x+3)^2(x-4)^3 \).
Solución
1. Since the polynomial has a zero at \(-3\) with multiplicity 2, one factor must be \((x+3)^2\).
2. Since the polynomial has a zero at \(4\) with multiplicity 3, another factor must be \((x-4)^3\).
3. The polynomial has a leading coefficient of \(2\), so the final expression for the polynomial must be multiplied by \(2\).
Thus, the polynomial in complete factored form is:
\[
f(x) = 2(x+3)^2(x-4)^3
\]
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Simplifique esta solución Extra Insights
To construct the polynomial \( f(x) \) with the given conditions, we start by expressing the zeros with their respective multiplicities. The zero at -3 has multiplicity 2, so its factor will be \( (x + 3)^2 \). The zero at 4 has multiplicity 3, leading to the factor \( (x - 4)^3 \). Putting it all together, we have the polynomial in factored form as follows: \[ f(x) = 2(x + 3)^2(x - 4)^3 \] This gives us a polynomial with a leading coefficient of 2, which satisfies all the given conditions.
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