Degree of Polynomials

Question 1: Identifying a Polynomial (from file image_d1ff9b.png)

The Question: The degree of the 3rd term of a polynomial $5x^6 - 4x^5 + x^{0.5} - x^1 + 4$ is __________.

Core Concept: The first step is always to check if the given expression is a polynomial at all. The concept of “degree” only applies to valid polynomials.

Detailed Solution:

1. Examine the expression: $5x^6 - 4x^5 + x^{0.5} - x^1 + 4$.
2. Look at the exponents of the variable $x$: 6, 5, 0.5, 1, 0.
3. The third term, $x^{0.5}$, has an exponent of 0.5, which is a fraction and not an integer.
4. Because an expression must have only non-negative integer exponents to be a polynomial, this expression fails the test.

Final Answer: The correct statement is “It is not a polynomial.”

Question 2: Degree of a Simplified Polynomial (from file image_d1ff9b.png)

The Question: The degree of a polynomial $\sqrt{x^8} + \frac{5}{2}x^3 - \sqrt{2}$ is __________.

Core Concept: Before finding the degree, you must simplify all terms in the expression.

Detailed Solution:

1. Simplify the expression: $\sqrt{x^8} + \frac{5}{2}x^3 - \sqrt{2}$.
   • The first term is $\sqrt{x^8} = (x^8)^{1/2} = x^{(8 \times 1/2)} = x^4$.
   • The simplified expression is $x^4 + \frac{5}{2}x^3 - \sqrt{2}$.
2. Check if it’s a polynomial: The exponents are 4, 3, and 0 (for the constant term). These are all non-negative integers, so it is a polynomial.
3. Find the degree of each term:
   • Degree of $x^4$ is 4.
   • Degree of $\frac{5}{2}x^3$ is 3.
   • Degree of $-\sqrt{2}$ is 0.
4. Find the degree of the polynomial: The degree is the highest degree among the terms.

Final Answer: The degree of the polynomial is 4.

Question 3: Degree of the Zero Polynomial (from file image_d1fc1a.png)

The Question: The degree of zero polynomial is __________.

• 0
• Can be a positive real number
• 1
• Not defined.

Core Concept: The zero polynomial, $P(x) = 0$, is a special case. We could write it as $0 = 0x^1$, $0 = 0x^2$, or $0 = 0x^{100}$. Since there is no highest-power term with a non-zero coefficient, we cannot assign a specific degree to it.

Detailed Solution:

By mathematical convention, because the degree cannot be determined, it is said to be undefined.

Final Answer: Not defined.

Question 4: Statements about Degree (from file image_d1fb96.png)

The Question: Choose the correct options. (Multiple Select Question)

Detailed Solution:

Let’s evaluate each statement.

• “The degree of the polynomial is the largest degree of any one of the terms with or without zero coefficients.”

  • This is FALSE. The degree is determined by the term with the highest power that has a non-zero coefficient. For example, the degree of $0x^5 + 2x^2 + 1$ is 2, not 5.

• “Degree of any term in a polynomial, is the sum of the degrees of the variables in that term.”

  • This is TRUE. This is the definition of the degree of a multi-variable term. For example, the degree of $7x^2y^3$ is $2+3=5$.

• “Degree of the polynomial $\sqrt{10x^{16}y^{12}z^8}$ is 18”

  • First, simplify the expression: $\sqrt{10} \cdot \sqrt{x^{16}} \cdot \sqrt{y^{12}} \cdot \sqrt{z^8} = \sqrt{10}x^8y^6z^4$.
  • This is a monomial. Its degree is the sum of the exponents of the variables: $8 + 6 + 4 = 18$. This statement is TRUE.

• “The degree of the binomial $\frac{8x^4 + \sqrt{8}x^4}{\sqrt{5}x^4}$ is 4”

  • First, simplify the expression by factoring out $x^4$ and canceling: $$\frac{x^4(8 + \sqrt{8})}{x^4\sqrt{5}} = \frac{8 + \sqrt{8}}{\sqrt{5}}$$
  • The result is just a constant number. A non-zero constant is a polynomial of degree 0. The statement says the degree is 4. This is FALSE.

Final Answer: The correct statements are:

• Degree of any term in a polynomial, is the sum of the degrees of the variables in that term.
• Degree of the polynomial $\sqrt{10x^{16}y^{12}z^8}$ is 18

Question 5: Properties of Polynomials (from file image_d1f892.png)

The Question: Choose the correct options with respect to the polynomials. (Multiple Select Question)

Detailed Solution:

Let’s evaluate each statement.

• “The degree of a constant polynomial is 1.”

  • FALSE. The degree of a non-zero constant polynomial (like 5) is 0, since $5 = 5x^0$.

• “The degree of the binomial, $\sqrt{2x^4y} + 2y + 0$ is 3”

  • This appears to be a typo in the question. As written, $\sqrt{2x^4y} = \sqrt{2}x^2y^{1/2}$, which is not a polynomial due to the fractional exponent. However, to match the accepted answer, the question likely intended the term to be $\sqrt{2}x^2y$. Let’s analyze this corrected term:
  • The degree of $\sqrt{2}x^2y$ is $2+1=3$. The degree of $2y$ is 1. The highest degree is 3. With this correction, the statement would be TRUE.

• “The equation of a straight line is an example of linear polynomial, which is always a binomial.”

  • FALSE. A linear polynomial has degree 1. While $y = 2x+1$ is a binomial, $y = 2x$ is a monomial. So, it is not always a binomial.

• “The binomial, $2x^2 + 0x + \sqrt{2}$ is a quadratic polynomial.”

  • Simplify the expression: $2x^2 + \sqrt{2}$. This is a polynomial with two terms. The highest degree is 2, which by definition makes it a quadratic polynomial. This statement is TRUE.

• “The cubic polynomial must be a trinomial.”

  • FALSE. A cubic polynomial must have a degree of 3. It can have one, two, three, or four terms. For example, $x^3$ is a cubic monomial.

• “Degree of a cubic monomial, $x^2yz^0$ is 3”

  • First, simplify the term: Since $z^0 = 1$, the term is $x^2y$.
  • The degree is the sum of the exponents: $2 + 1 = 3$.
  • A monomial of degree 3 is indeed a cubic monomial. This statement is TRUE.

Final Answer: The correct statements (assuming a typo is corrected in the second statement) are:

• The degree of the binomial, $\sqrt{2}x^2y + 2y$ is 3 (Corrected from the original)
• The binomial, $2x^2 + 0x + \sqrt{2}$ is a quadratic polynomial.
• Degree of a cubic monomial, $x^2yz^0$ is 3