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Answer :
- Find all zeros of the polynomial, including complex conjugates: $5, 2, 3-3i, 3+3i$.
- Multiply the factors corresponding to the zeros: $(x-5)(x-2)(x-(3-3i))(x-(3+3i))$.
- Simplify the expression: $(x^2 - 7x + 10)(x^2 - 6x + 18)$.
- Obtain the polynomial: $x^4 - 13x^3 + 70x^2 - 186x + 180$. The final answer is $\boxed{x^4-13 x^3+70 x^2-186 x+180}$
### Explanation
1. Identifying all Zeros
We are given the zeros of a polynomial as $5, 2, 3-3i$. We need to find the polynomial function of least degree with integral coefficients. Since the polynomial has integral coefficients, if a complex number is a zero, then its complex conjugate is also a zero. Thus, if $3-3i$ is a zero, then $3+3i$ is also a zero. Therefore, the zeros are $5, 2, 3-3i, 3+3i$.
2. Building the Polynomial
Since the zeros are $5, 2, 3-3i, 3+3i$, the polynomial can be written as $f(x) = (x-5)(x-2)(x-(3-3i))(x-(3+3i))$. First, we multiply $(x-5)(x-2)$ to get $x^2 - 2x - 5x + 10 = x^2 - 7x + 10$.
3. Multiplying Complex Factors
Next, we multiply $(x-(3-3i))(x-(3+3i))$ to get $(x-3+3i)(x-3-3i)$. This can be rewritten as $((x-3)+3i)((x-3)-3i)$. Using the difference of squares formula, we have $(x-3)^2 - (3i)^2 = x^2 - 6x + 9 - (9i^2) = x^2 - 6x + 9 - (-9) = x^2 - 6x + 18$.
4. Final Polynomial
Finally, we multiply $(x^2 - 7x + 10)(x^2 - 6x + 18)$ to get the polynomial function. Expanding the expression: $f(x) = (x^2 - 7x + 10)(x^2 - 6x + 18) = x^4 - 6x^3 + 18x^2 - 7x^3 + 42x^2 - 126x + 10x^2 - 60x + 180 = x^4 - 13x^3 + 70x^2 - 186x + 180$.
5. Conclusion
Therefore, the polynomial function of least degree with integral coefficients that has the given zeros is $f(x) = x^4 - 13x^3 + 70x^2 - 186x + 180$.
### Examples
Polynomial functions are used in various fields such as physics, engineering, and economics to model complex relationships. For example, in physics, projectile motion can be modeled using a quadratic polynomial. In economics, cost and revenue functions can be represented using polynomials to analyze business performance. Understanding how to construct polynomials with specific roots is crucial for creating accurate models in these fields.
- Multiply the factors corresponding to the zeros: $(x-5)(x-2)(x-(3-3i))(x-(3+3i))$.
- Simplify the expression: $(x^2 - 7x + 10)(x^2 - 6x + 18)$.
- Obtain the polynomial: $x^4 - 13x^3 + 70x^2 - 186x + 180$. The final answer is $\boxed{x^4-13 x^3+70 x^2-186 x+180}$
### Explanation
1. Identifying all Zeros
We are given the zeros of a polynomial as $5, 2, 3-3i$. We need to find the polynomial function of least degree with integral coefficients. Since the polynomial has integral coefficients, if a complex number is a zero, then its complex conjugate is also a zero. Thus, if $3-3i$ is a zero, then $3+3i$ is also a zero. Therefore, the zeros are $5, 2, 3-3i, 3+3i$.
2. Building the Polynomial
Since the zeros are $5, 2, 3-3i, 3+3i$, the polynomial can be written as $f(x) = (x-5)(x-2)(x-(3-3i))(x-(3+3i))$. First, we multiply $(x-5)(x-2)$ to get $x^2 - 2x - 5x + 10 = x^2 - 7x + 10$.
3. Multiplying Complex Factors
Next, we multiply $(x-(3-3i))(x-(3+3i))$ to get $(x-3+3i)(x-3-3i)$. This can be rewritten as $((x-3)+3i)((x-3)-3i)$. Using the difference of squares formula, we have $(x-3)^2 - (3i)^2 = x^2 - 6x + 9 - (9i^2) = x^2 - 6x + 9 - (-9) = x^2 - 6x + 18$.
4. Final Polynomial
Finally, we multiply $(x^2 - 7x + 10)(x^2 - 6x + 18)$ to get the polynomial function. Expanding the expression: $f(x) = (x^2 - 7x + 10)(x^2 - 6x + 18) = x^4 - 6x^3 + 18x^2 - 7x^3 + 42x^2 - 126x + 10x^2 - 60x + 180 = x^4 - 13x^3 + 70x^2 - 186x + 180$.
5. Conclusion
Therefore, the polynomial function of least degree with integral coefficients that has the given zeros is $f(x) = x^4 - 13x^3 + 70x^2 - 186x + 180$.
### Examples
Polynomial functions are used in various fields such as physics, engineering, and economics to model complex relationships. For example, in physics, projectile motion can be modeled using a quadratic polynomial. In economics, cost and revenue functions can be represented using polynomials to analyze business performance. Understanding how to construct polynomials with specific roots is crucial for creating accurate models in these fields.
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