Let's break down each polynomial and classify them.

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9. $2m^3 - 3m^2 + 25m - 9$

• Standard Form: This polynomial is already in standard form, where the terms are ordered by the powers of $m$ in descending order.

  Classification:

  • Degree: 3 (The highest power of $m$ is 3)
  • Leading Term: $2m^3$
  • Type: Cubic Polynomial

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10. $k^2 - 3k + 3k^2 + 5$

• Simplify: Combine like terms: $k^2 + 3k^2 - 3k + 5 = 4k^2 - 3k + 5$ Now it's in standard form.

  Classification:

  • Degree: 2 (The highest power of $k$ is 2)
  • Leading Term: $4k^2$
  • Type: Quadratic Polynomial

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11. $2y^2 - 3x^2 + 14xy$

• Standard Form: The terms are already written with the powers of variables in descending order (considering both $x$ and $y$).

  Classification:

  • Degree: 3 (The highest combined degree is 2 for $x^2$ and 1 for $xy$, which gives 3 for the $14xy$ term.)
  • Leading Term: $14xy$
  • Type: Cubic Polynomial (In two variables)

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12. $9x$

• Standard Form: This is already in standard form.

  Classification:

  • Degree: 1 (The highest power of $x$ is 1)
  • Leading Term: $9x$
  • Type: Linear Polynomial

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Summary of Classifications:

1. $2m^3 - 3m^2 + 25m - 9$: Cubic Polynomial
2. $k^2 - 3k + 3k^2 + 5$: Quadratic Polynomial
3. $2y^2 - 3x^2 + 14xy$: Cubic Polynomial (in two variables)
4. $9x$: Linear Polynomial

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Do you need any further clarification on these steps, or would you like more details?

Here are some additional questions that relate to what we've covered:

1. How do you identify the degree of a polynomial in two variables?
2. What is the difference between a monomial, binomial, and trinomial?
3. How can you combine like terms in polynomials with multiple variables?
4. What happens if you have negative exponents in a polynomial?
5. What is the general method to classify polynomials by their degree?

Tip: When classifying polynomials, always start by writing them in standard form with the terms ordered by degree, from the highest to the lowest.