To write the polynomial function in standard form that has a leading coefficient of 1 with the least degree for the zeros $1$, $-3$, and $\sqrt{5}$, we follow these steps:

Step 1: Write the factors for each zero.

• For $x = 1$, the factor is $(x - 1)$.
• For $x = -3$, the factor is $(x + 3)$.
• For $x = \sqrt{5}$, the factor is $(x - \sqrt{5})$.

Step 2: Include the conjugate of $\sqrt{5}$.

Since polynomials with real coefficients cannot have an irrational number as a single zero without its conjugate, we must also include $-\sqrt{5}$ as a zero. Thus, the corresponding factor is $(x + \sqrt{5})$.

Step 3: Write the polynomial.

Now, the polynomial is the product of all the factors: $f(x) = (x - 1)(x + 3)(x - \sqrt{5})(x + \sqrt{5})$

Step 4: Simplify the expression.

First, simplify the product of the conjugate pair $(x - \sqrt{5})(x + \sqrt{5})$: $(x - \sqrt{5})(x + \sqrt{5}) = x^2 - (\sqrt{5})^2 = x^2 - 5$

Now the polynomial becomes: $f(x) = (x - 1)(x + 3)(x^2 - 5)$

Next, expand $(x - 1)(x + 3)$: $(x - 1)(x + 3) = x^2 + 3x - x - 3 = x^2 + 2x - 3$

Now the polynomial is: $f(x) = (x^2 + 2x - 3)(x^2 - 5)$

Finally, expand this product: $f(x) = x^2(x^2 - 5) + 2x(x^2 - 5) - 3(x^2 - 5)$ $f(x) = x^4 - 5x^2 + 2x^3 - 10x - 3x^2 + 15$ $f(x) = x^4 + 2x^3 - 8x^2 - 10x + 15$

Step 5: Final polynomial in standard form.

The polynomial in standard form, with a leading coefficient of 1, is: $f(x) = x^4 + 2x^3 - 8x^2 - 10x + 15$

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Would you like further details or have any questions about this process?

Here are five related questions to explore:

1. How do you find the zeros of a polynomial given its factors?
2. What is the importance of including conjugate pairs when working with irrational zeros?
3. How do you determine the degree of a polynomial?
4. Can a polynomial with complex roots have real coefficients?
5. What does the standard form of a polynomial tell you about its graph?

Tip: Always check if the polynomial has real coefficients and include conjugates when necessary!