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Answer :
- Identify all roots: Given roots are $\frac{5}{3}$ and $-3i$, and since the polynomial has integral coefficients, $3i$ is also a root.
- Write the factors: The factors are $(x - \frac{5}{3})$, $(x + 3i)$, and $(x - 3i)$.
- Multiply complex conjugate factors: $(x + 3i)(x - 3i) = x^2 + 9$.
- Multiply remaining factors and adjust for integral coefficients: $3(x - \frac{5}{3})(x^2 + 9) = 3x^3 - 5x^2 + 27x - 45$.
- The polynomial function is $\boxed{3 x^3-5 x^2+27 x-45}$.
### Explanation
1. Identify all roots
We are given the zeros $\frac{5}{3}$ and $-3i$ of a polynomial function with integral coefficients. Since the coefficients are integers, complex roots must come in conjugate pairs. Thus, if $-3i$ is a root, then $3i$ must also be a root.
2. Write the factors
The roots are $\frac{5}{3}, -3i,$ and $3i$. Therefore, the factors of the polynomial are $(x - \frac{5}{3})$, $(x + 3i)$, and $(x - 3i)$.
3. Multiply complex conjugate factors
First, multiply the factors corresponding to the complex conjugate roots: $(x + 3i)(x - 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9$.
4. Multiply remaining factors
Next, multiply the remaining factor: $(x - \frac{5}{3})(x^2 + 9) = x^3 + 9x - \frac{5}{3}x^2 - \frac{5}{3}(9) = x^3 - \frac{5}{3}x^2 + 9x - 15$.
5. Adjust for integral coefficients
To obtain integral coefficients, multiply the polynomial by 3: $3(x^3 - \frac{5}{3}x^2 + 9x - 15) = 3x^3 - 5x^2 + 27x - 45$.
6. Final polynomial
Therefore, the polynomial function of least degree with integral coefficients is $f(x) = 3x^3 - 5x^2 + 27x - 45$.
### Examples
Polynomial functions are used in various fields such as physics, engineering, and economics. For example, they can model the trajectory of a projectile, the shape of a bridge, or the growth of a population. Understanding how to construct a polynomial function from its zeros is essential for solving real-world problems in these areas. In computer graphics, polynomials are used to create curves and surfaces. In control systems, they are used to design stable systems.
- Write the factors: The factors are $(x - \frac{5}{3})$, $(x + 3i)$, and $(x - 3i)$.
- Multiply complex conjugate factors: $(x + 3i)(x - 3i) = x^2 + 9$.
- Multiply remaining factors and adjust for integral coefficients: $3(x - \frac{5}{3})(x^2 + 9) = 3x^3 - 5x^2 + 27x - 45$.
- The polynomial function is $\boxed{3 x^3-5 x^2+27 x-45}$.
### Explanation
1. Identify all roots
We are given the zeros $\frac{5}{3}$ and $-3i$ of a polynomial function with integral coefficients. Since the coefficients are integers, complex roots must come in conjugate pairs. Thus, if $-3i$ is a root, then $3i$ must also be a root.
2. Write the factors
The roots are $\frac{5}{3}, -3i,$ and $3i$. Therefore, the factors of the polynomial are $(x - \frac{5}{3})$, $(x + 3i)$, and $(x - 3i)$.
3. Multiply complex conjugate factors
First, multiply the factors corresponding to the complex conjugate roots: $(x + 3i)(x - 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9$.
4. Multiply remaining factors
Next, multiply the remaining factor: $(x - \frac{5}{3})(x^2 + 9) = x^3 + 9x - \frac{5}{3}x^2 - \frac{5}{3}(9) = x^3 - \frac{5}{3}x^2 + 9x - 15$.
5. Adjust for integral coefficients
To obtain integral coefficients, multiply the polynomial by 3: $3(x^3 - \frac{5}{3}x^2 + 9x - 15) = 3x^3 - 5x^2 + 27x - 45$.
6. Final polynomial
Therefore, the polynomial function of least degree with integral coefficients is $f(x) = 3x^3 - 5x^2 + 27x - 45$.
### Examples
Polynomial functions are used in various fields such as physics, engineering, and economics. For example, they can model the trajectory of a projectile, the shape of a bridge, or the growth of a population. Understanding how to construct a polynomial function from its zeros is essential for solving real-world problems in these areas. In computer graphics, polynomials are used to create curves and surfaces. In control systems, they are used to design stable systems.
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