Write a polynomial f(x) that satisfies the given conditions. Polynomial of lowest degree with zeros of (5)/(3) (multiplicity 2 ) and -(2)/(3) (multiplicity 1 ) and with f(0)=-100.

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Write a polynomial  f(x) that satisfies the given conditions. Polynomial of lowest degree with zeros of  (5)/(3) (multiplicity 2 ) and  -(2)/(3) (multiplicity 1 ) and with  f(0)=-100.

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Identify Zeros and Factors: Identify the linear factors associated with the given zeros, considering their multiplicities. The zero (5/3) with multiplicity 2 gives us the factor (x - 5/3)^2, and the zero -(2/3) with multiplicity 1 gives us the factor (x + 2/3).Write Factored Form: Write the polynomial in its factored form using the identified factors. The polynomial is f(x) = (x - 5/3)^2 * (x + 2/3).Expand and Simplify: Expand the polynomial to find the standard form. First, expand (x - 5/3)^2 to get x^2 - (2*5/3)x + (5/3)^2 = x^2 - (10/3)x + 25/9.Distribute Terms: Next, expand the entire polynomial f(x) = (x^2 - (10/3)x + 25/9) * (x + 2/3).Combine Like Terms: Distribute the terms to get f(x) = x^3 + (2/3)x^2 - (10/3)x^2 - (20/9)x + (25/9)x + (50/27).Adjust Constant Term: Combine like terms to simplify the polynomial. We get f(x) = x^3 - (8/3)x^2 + (5/9)x + (50/27).Calculate New Constant: Now, we need to adjust the polynomial so that f(0) = -100. Substitute x = 0 into the polynomial to find the current value of f(0). We get f(0) = 0^3 - (8/3)*0^2 + (5/9)*0 + (50/27) = 50/27.Write Final Polynomial: To make f(0) = -100, we need to adjust the constant term. Since f(0) is currently 50/27, we need to subtract (50/27 + 100) from the constant term to get the desired value. Calculate the new constant term: (50/27) - (50/27 + 100) = -100.Write the final polynomial with the adjusted constant term. The polynomial is f(x) = x^3 - (8/3)x^2 + (5/9)x - 100.

Write a polynomial $f(x)$ that satisfies the given conditions. Polynomial of lowest degree with zeros of $\frac{5}{3}$ (multiplicity $2$ ) and $-\frac{2}{3}$ (multiplicity $1$ ) and with $f(0)=-100$ .

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Write a polynomial $f(x)$ that satisfies the given conditions. Polynomial of lowest degree with zeros of $\frac{5}{3}$ (multiplicity $2$ ) and $-\frac{2}{3}$ (multiplicity $1$ ) and with $f(0)=-100$ .