To construct a polynomial with the given zeros $-3$, $1$, and $3 + \sqrt{2}$, we must ensure the polynomial is in standard form and has real coefficients. Since $3 + \sqrt{2}$ is an irrational root, its conjugate $3 - \sqrt{2}$ must also be a root for the polynomial to have real coefficients.

Step 1: Represent the polynomial as a product of factors

For each root $r$, the corresponding factor is $(x - r)$. Thus, the polynomial can be expressed as: $P(x) = (x + 3)(x - 1)\big(x - (3 + \sqrt{2})\big)\big(x - (3 - \sqrt{2})\big).$

Step 2: Simplify the factors

The product involving the irrational roots can be simplified using the difference of squares: $\big(x - (3 + \sqrt{2})\big)\big(x - (3 - \sqrt{2})\big) = \big(x - 3\big)^2 - (\sqrt{2})^2 = \big(x - 3\big)^2 - 2.$

Step 3: Write the polynomial

Substituting this back, we have: $P(x) = (x + 3)(x - 1)\big[\big(x - 3\big)^2 - 2\big].$

Step 4: Expand the terms

1. Expand $\big(x - 3\big)^2 - 2$: $\big(x - 3\big)^2 = x^2 - 6x + 9, \quad \text{so } \big(x - 3\big)^2 - 2 = x^2 - 6x + 7.$

2. Substitute back into $P(x)$: $P(x) = (x + 3)(x - 1)(x^2 - 6x + 7).$

3. Expand $(x + 3)(x - 1)$: $(x + 3)(x - 1) = x^2 + 2x - 3.$

4. Multiply $(x^2 + 2x - 3)$ by $(x^2 - 6x + 7)$: $P(x) = (x^2 + 2x - 3)(x^2 - 6x + 7).$ Expand this step by step: $x^2(x^2 - 6x + 7) = x^4 - 6x^3 + 7x^2,$ $2x(x^2 - 6x + 7) = 2x^3 - 12x^2 + 14x,$ $-3(x^2 - 6x + 7) = -3x^2 + 18x - 21.$ Combine terms: $P(x) = x^4 - 6x^3 + 7x^2 + 2x^3 - 12x^2 + 14x - 3x^2 + 18x - 21.$ $P(x) = x^4 - 4x^3 - 8x^2 + 32x - 21.$

Final Answer

The polynomial in standard form is: $P(x) = x^4 - 4x^3 - 8x^2 + 32x - 21.$

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Would you like a step-by-step walkthrough of any part?

Here are five related questions to deepen your understanding:

1. How do you verify that a polynomial has the given roots?
2. Why must irrational roots appear in conjugate pairs in polynomials with real coefficients?
3. What is the degree of a polynomial determined by its roots?
4. How can the standard form of a polynomial be checked for accuracy?
5. How does the structure of a polynomial reflect its zeros?

Tip: When expanding polynomials, keep track of terms by organizing work step by step to avoid errors.