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Find the polynomial function
f with real coefficients that has the given degree, zeros, and solution point.

{:[" Degree "," Zeros "," Solution Point "],[3,-9","1+sqrt3i,f(-2)=84]:}

Find the polynomial function $f$ with real coefficients that has the given degree, zeros, and solution point. $\newline$ $\begin{array}{ccc} \text { Degree } & \text { Zeros } & \text { Solution Point } \\ 3 & -9,1+\sqrt{3} i & f(-2)=84 \end{array}$

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Write a polynomial from its roots

Full solution

Q. Find the polynomial function

f

with real coefficients that has the given degree, zeros, and solution point.

\newline

\begin{array}{ccc} \text { Degree } & \text { Zeros } & \text { Solution Point } \\ 3 & -9,1+\sqrt{3} i & f(-2)=84 \end{array}

Apply Complex Conjugate Root Theorem: Since the polynomial has real coefficients and one of the zeros is a complex number $(1+\sqrt{3}i)$ , the complex conjugate $(1-\sqrt{3}i)$ must also be a zero of the polynomial due to the Complex Conjugate Root Theorem.
Factorize the Polynomial: The polynomial must have the factors $(x + 9)$ , $(x - (1+\sqrt{3}i))$ , and $(x - (1-\sqrt{3}i))$ to account for the zeros $-9$ , $1+\sqrt{3}i$ , and $1-\sqrt{3}i$ , respectively.
Multiply Complex Factors: We can write the polynomial in its factored form as $f(x) = (x + 9)(x - (1+\sqrt{3}i))(x - (1-\sqrt{3}i))$ .
Expand and Simplify: To find the polynomial in standard form, we first multiply the factors involving the complex zeros: $(x - (1+\sqrt{3}i))(x - (1-\sqrt{3}i))$ .
Determine Leading Coefficient: Multiplying the complex factors, we get: $(x - 1 - \sqrt{3}i)(x - 1 + \sqrt{3}i) = (x - 1)^2 - (\sqrt{3}i)^2 = x^2 - 2x + 1 - 3 = x^2 - 2x - 2$ .
Substitute Point $(-2, 84)$ : Now we multiply this result by the remaining factor $(x + 9)$ to get the polynomial in standard form: $f(x) = (x + 9)(x^2 - 2x - 2)$ .
Calculate Adjusted Value: Expanding the polynomial, we get: $f(x) = x^3 - 2x^2 - 2x + 9x^2 - 18x - 18 = x^3 + 7x^2 - 20x - 18$ .
Final Polynomial: We now need to determine the leading coefficient. The problem states that the polynomial must pass through the point $(-2, 84)$ , so we substitute $x = -2$ into the polynomial and set it equal to $84$ : $f(-2) = (-2)^3 + 7(-2)^2 - 20(-2) - 18 = 84$ .
Final Polynomial: We now need to determine the leading coefficient. The problem states that the polynomial must pass through the point $(-2, 84)$ , so we substitute $x = -2$ into the polynomial and set it equal to $84$ : $f(-2) = (-2)^3 + 7(-2)^2 - 20(-2) - 18 = 84$ . Calculating the value of $f(-2)$ , we get: $f(-2) = -8 + 7(4) + 40 - 18 = -8 + 28 + 40 - 18 = 42$ .
Final Polynomial: We now need to determine the leading coefficient. The problem states that the polynomial must pass through the point $(-2, 84)$ , so we substitute $x = -2$ into the polynomial and set it equal to $84$ : $f(-2) = (-2)^3 + 7(-2)^2 - 20(-2) - 18 = 84$ . Calculating the value of $f(-2)$ , we get: $f(-2) = -8 + 7(4) + 40 - 18 = -8 + 28 + 40 - 18 = 42$ . Since $f(-2) = 42$ , not $84$ , we need to find a constant $k$ such that $k f(-2) = 84$ . We solve for $k$ : $x = -2$ $1$ , so $x = -2$ $2$ .
Final Polynomial: We now need to determine the leading coefficient. The problem states that the polynomial must pass through the point $(-2, 84)$ , so we substitute $x = -2$ into the polynomial and set it equal to $84$ : $f(-2) = (-2)^3 + 7(-2)^2 - 20(-2) - 18 = 84$ . Calculating the value of $f(-2)$ , we get: $f(-2) = -8 + 7(4) + 40 - 18 = -8 + 28 + 40 - 18 = 42$ . Since $f(-2) = 42$ , not $84$ , we need to find a constant $k$ such that $kf(-2) = 84$ . We solve for $k$ : $x = -2$ $1$ , so $x = -2$ $2$ . We multiply the entire polynomial by $k$ to get the final polynomial: $x = -2$ $4$ .

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