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Question 1: Determine if the expression is a polynomial, and if it is, state the degree. If it is not, state why.

• a. $2x^2 - 3x + 5$
• b. $\sqrt{x} = 7$
• c. $3x^3 + 7x^2 - 4y$
• d. $a^3 - 11$

Question 2: Simplify as much as possible.

• a. $(6x^2 + 5x + 10) + (4x^2 + 6x + 12)$
• b. $(2x^2 - 3) - (5x + 6)$
• c. $(x^2 + 2x - 5) + (3x^2 - 4x + 7)$
• d. $(7x^4 - 5x^3 + 8x^2 + 9x - 13) - (2x^4 + 6x^3 - 4x^2 - x + 4)$

Let's address the first question on polynomials:

1. a. $2x^2 - 3x + 5$: This is a polynomial. The degree is 2 because the highest exponent is 2.
2. b. $\sqrt{x} = 7$: This is not a polynomial because the expression involves a square root, which is not allowed in polynomials.
3. c. $3x^3 + 7x^2 - 4y$: This is a polynomial in terms of $x$, but since it involves $y$, it is considered a multi-variable polynomial. The degree is 3 because the highest power of $x$ is 3.
4. d. $a^3 - 11$: This is a polynomial in $a$, and the degree is 3, as the highest power of $a$ is 3.

Now, simplifying the expressions:

• a. For $(6x^2 + 5x + 10) + (4x^2 + 6x + 12)$, combine like terms:

  • $6x^2 + 4x^2 = 10x^2$
  • $5x + 6x = 11x$
  • $10 + 12 = 22$
  • Final result: $10x^2 + 11x + 22$

• b. For $(2x^2 - 3) - (5x + 6)$, distribute the negative sign and combine like terms:

  • $2x^2 - 3 - 5x - 6 = 2x^2 - 5x - 9$
  • Final result: $2x^2 - 5x - 9$

• c. For $(x^2 + 2x - 5) + (3x^2 - 4x + 7)$, combine like terms:

  • $x^2 + 3x^2 = 4x^2$
  • $2x - 4x = -2x$
  • $-5 + 7 = 2$
  • Final result: $4x^2 - 2x + 2$

• d. For $(7x^4 - 5x^3 + 8x^2 + 9x - 13) - (2x^4 + 6x^3 - 4x^2 - x + 4)$, distribute the negative sign and combine like terms:

  • $7x^4 - 2x^4 = 5x^4$
  • $-5x^3 - 6x^3 = -11x^3$
  • $8x^2 + 4x^2 = 12x^2$
  • $9x + x = 10x$
  • $-13 - 4 = -17$
  • Final result: $5x^4 - 11x^3 + 12x^2 + 10x - 17$

Let me know if you'd like further clarification on any of these steps or if you have any other questions!

5 Follow-up Questions:

1. How do you identify polynomials in multiple variables?
2. How would you determine the degree of a polynomial with fractions or negative exponents?
3. Can polynomials have terms with negative coefficients?
4. What is the difference between polynomials and rational expressions?
5. How do you perform polynomial division?

Tip:

When combining like terms, always make sure the powers of the variables are the same before adding or subtracting their coefficients.