https://file.aiquickdraw.com/mathbot/file/1729006337134i9hx54vl.jpg

Let's break down the questions from the image and work on the solutions step by step:

### 1. Polynomial function with rational coefficients (Zeros: \(2, -1, \sqrt{3}\))
Since we are asked to create a polynomial with rational coefficients, we must also include the conjugate of \(\sqrt{3}\), which is \(-\sqrt{3}\). Thus, the zeros are: \(2, -1, \sqrt{3}, -\sqrt{3}\).

The polynomial is formed by creating factors for each zero:
\[
f(x) = (x - 2)(x + 1)(x - \sqrt{3})(x + \sqrt{3})
\]

Simplify the factors with square roots:
\[
f(x) = (x - 2)(x + 1)((x - \sqrt{3})(x + \sqrt{3})) = (x - 2)(x + 1)(x^2 - 3)
\]

Now, expand the first part:
\[
(x - 2)(x + 1) = x^2 - x - 2
\]

Finally, expand everything:
\[
f(x) = (x^2 - x - 2)(x^2 - 3) = x^4 - 3x^2 - x^3 + 3x - 2x^2 + 6
\]
\[
f(x) = x^4 - x^3 - 5x^2 + 3x + 6
\]

So, the polynomial is:
\[
f(x) = x^4 - x^3 - 5x^2 + 3x + 6
\]

### 2. Polynomial function with rational coefficients (Zeros: \(3, -2, 2i\))
Since the zeros include \(2i\), we also need to include its conjugate \(-2i\). The zeros are \(3, -2, 2i, -2i\).

The polynomial is:
\[
f(x) = (x - 3)(x + 2)(x - 2i)(x + 2i)
\]

Simplify the imaginary part:
\[
(x - 2i)(x + 2i) = x^2 + 4
\]

Now expand the remaining factors:
\[
(x - 3)(x + 2) = x^2 - x - 6
\]

Finally, expand everything:
\[
f(x) = (x^2 - x - 6)(x^2 + 4) = x^4 + 4x^2 - x^3 - 4x - 6x^2 - 24
\]
\[
f(x) = x^4 - x^3 - 2x^2 - 4x - 24
\]

So, the polynomial is:
\[
f(x) = x^4 - x^3 - 2x^2 - 4x - 24
\]

### 3. Find the product or quotient: \(\frac{3x^3 + 6x^2 - 5x + 12}{x - 3}\)
Perform synthetic or polynomial division here.

Using synthetic division with \(x = 3\):

1. Coefficients of the polynomial: \(3, 6, -5, 12\)
2. Synthetic division by \(3\):

\[
\begin{array}{r|rrrr}
3 & 3 & 6 & -5 & 12 \\
  &   & 9 & 45 & 120 \\
\hline
  & 3 & 15 & 40 & 132
\end{array}
\]

The quotient is:
\[
3x^2 + 15x + 40 \quad \text{(remainder = 132)}
\]

So the result is:
\[
3x^2 + 15x + 40 + \frac{132}{x-3}
\]

### 4. Expand: \((3x - 2)^3\)
We use the binomial expansion for \((a - b)^3\):

\[
(3x - 2)^3 = (3x)^3 - 3(3x)^2(2) + 3(3x)(2^2) - 2^3
\]
\[
= 27x^3 - 54x^2 + 36x - 8
\]

### 5. Expand: \((m - 2)(m + 3)(m - 1)\)
First, multiply \((m - 2)(m + 3)\):
\[
(m - 2)(m + 3) = m^2 + 3m - 2m - 6 = m^2 + m - 6
\]

Now multiply this by \((m - 1)\):
\[
(m^2 + m - 6)(m - 1) = m^3 - m^2 + m^2 - m - 6m + 6 = m^3 - 7m + 6
\]

### 6. Divide: \(\frac{2x^4 - x^2 + 3x + 1}{x^2 + 2x + 2}\)
Perform polynomial long division.

1. Divide \(2x^4\) by \(x^2\) to get \(2x^2\).
2. Multiply \((x^2 + 2x + 2)\) by \(2x^2\) and subtract from the original polynomial.
3. Repeat the process until reaching the remainder.

The final result would be in the form \(Q(x) + \frac{R(x)}{D(x)}\), where \(Q(x)\) is the quotient and \(R(x)\) is the remainder.

---

Would you like further explanation or clarifications on any of these steps?

### 5 Related Questions:
1. How do we derive the conjugate of irrational numbers in polynomials?
2. What is the importance of rational coefficients in polynomial functions?
3. Can you explain the difference between synthetic and long division in polynomials?
4. How can we use the binomial theorem for higher powers of binomials?
5. What are the advantages of factoring polynomials before solving equations?

### Tip:
When working with complex or irrational roots, always include their conjugates to ensure that the polynomial has real coefficients.

Write a polynomial function f of least degree that has rational coefficients, leading coefficient of 1, and the given zeros. Also, find the product or quotient of the given expressions.

Learn how to find the polynomial of least degree with rational coefficients given specific roots, and simplify polynomial expressions using division and binomial expansion. Detailed solutions for high school math.

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