Questions: Find the polynomial function of lowest degree with only real coefficients and having the zeros sqrt(3), -sqrt(3), and 5. Choose the correct polynomial function of lowest degree with only real coefficients and having the zeros sqrt(3), -sqrt(3), and 5. A. f(x)=3 x^3+8 x^2-3 x-3 B. f(x)=x^4-7 x^3-8 x^2+5 x+9 C. f(x)=x^3-3 x^2-5 x+15 D. f(x)=x^3-5 x^2-3 x+15

Find the polynomial function of lowest degree with only real coefficients and having the zeros sqrt(3), -sqrt(3), and 5. Choose the correct polynomial function of lowest degree with only real coefficients and having the zeros sqrt(3), -sqrt(3), and 5. A. f(x)=3 x^3+8 x^2-3 x-3 B. f(x)=x^4-7 x^3-8 x^2+5 x+9 C. f(x)=x^3-3 x^2-5 x+15 D. f(x)=x^3-5 x^2-3 x+15
Transcript text: Find the polynomial function of lowest degree with only real coefficients and having the zeros $\sqrt{3},-\sqrt{3}$, and 5 . Choose the correct polynomial function of lowest degree with only real coefficients and having the zeros $\sqrt{3},-\sqrt{3}$, and 5 . A. $f(x)=3 x^{3}+8 x^{2}-3 x-3$ B. $f(x)=x^{4}-7 x^{3}-8 x^{2}+5 x+9$ C. $f(x)=x^{3}-3 x^{2}-5 x+15$ D. $f(x)=x^{3}-5 x^{2}-3 x+15$
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Solution

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Solution Steps

Step 1: Pair Complex and Irrational Zeros

Given zeros are $\left[ \sqrt{3}, \ - \sqrt{3}, \ 5\right]$. Since the polynomial must have real (or rational) coefficients, any complex or irrational zeros are assumed to come in conjugate pairs.

Step 2: Form Factors for Each Zero

For each zero, a factor of the form $(x - z_i)$ is formed, resulting in factors: [$x - \sqrt{3}$,$x + \sqrt{3}$,$x - 5$,].

Step 3: Multiply Factors

Multiplying all the factors to get the polynomial: $f(x) = x^{3} - 5 x^{2} - 3 x + 15$.

Step 4: Simplify

The polynomial in its simplified standard form is $f(x) = x^{3} - 5 x^{2} - 3 x + 15$.

Final Answer:

The polynomial function of the lowest degree with real (or rational) coefficients, given the set of zeros $\left[ \sqrt{3}, \ - \sqrt{3}, \ 5\right]$, is $f(x) = x^{3} - 5 x^{2} - 3 x + 15$.

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