RBSE Solutions Class 9 Maths Chapter 3 Polynomial Exercise 3.1

Chapter 3 Polynomial Ex 3.1

Question 1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answers.
(i) \( 3x^2 - 5x + 13 \)
(ii) \( y^2 + 2\sqrt{3} \)
(iii) \( y + \frac {3}{y} \)
(iv) 3
(v) \( 2\sqrt{x} + \sqrt{3}x \)
(vi) \( x^{12} + y^3 + t^{20} \)
Answer:
(i) \( 3x^2 - 5x + 13 \) is a polynomial in one variable. This is because the power of the variable x (which is 2 and 1) is always a whole number. A polynomial requires all variable powers to be non-negative integers.
(ii) \( y^2 + 2\sqrt{3} \) is a polynomial in one variable. The power of the variable y is a whole number (2). The term \( 2\sqrt{3} \) is a constant and does not affect its polynomial status.
(iii) \( y + \frac {3}{y} \) is not a polynomial. This is because the term \( \frac {3}{y} \) can be written as \( 3y^{-1} \). The power of the variable y in this term is -1, which is not a whole number.
(iv) 3 is a constant polynomial. The degree of a constant polynomial is always 0. It can be seen as \( 3x^0 \).
(v) \( 2\sqrt{x} + \sqrt{3}x \) is not a polynomial. The term \( 2\sqrt{x} \) can be written as \( 2x^{\frac{1}{2}} \). The power of the variable x in this term is \( \frac{1}{2} \), which is not a whole number.
(vi) \( x^{12} + y^3 + t^{20} \) is not a polynomial in one variable. This expression has three different variables: x, y, and t. A polynomial in one variable must only contain one type of variable.
In simple words: To be a polynomial in one variable, all the powers of that single letter (like x or y) must be whole numbers (0, 1, 2, 3, etc.). If you see fractions or negative numbers as powers, or more than one letter, it's not a polynomial in one variable.

🎯 Exam Tip: Remember that for an expression to be a polynomial, the exponents of all variables must be non-negative integers. For a "polynomial in one variable," only one letter (like x or y) should be present in the terms.

Question 2. Write the coefficient of \( x^2 \) in each of the following:
(i) \( 12 + 3x + 5x^2 \)
(ii) \( 7 - 11x + x^3 \)
(iii) \( \sqrt{3}x - 7 \)
(iv) \( \frac {\pi }{2}{x}^{2}+x \)
Answer:
(i) In the polynomial \( 12 + 3x + 5x^2 \), the term with \( x^2 \) is \( 5x^2 \). The number multiplied by \( x^2 \) is 5. So, the coefficient of \( x^2 \) is 5.
(ii) In the polynomial \( 7 - 11x + x^3 \), there is no \( x^2 \) term visible. When a term is not present, its coefficient is considered to be 0. So, the coefficient of \( x^2 \) is 0.
(iii) In the polynomial \( \sqrt{3}x - 7 \), there is no \( x^2 \) term. So, the coefficient of \( x^2 \) is 0.
(iv) In the polynomial \( \frac {\pi }{2}{x}^{2}+x \), the term with \( x^2 \) is \( \frac {\pi }{2}{x}^{2} \). The number multiplied by \( x^2 \) is \( \frac {\pi }{2} \). So, the coefficient of \( x^2 \) is \( \frac {\pi }{2} \).
In simple words: The coefficient of \( x^2 \) is just the number that is right in front of the \( x^2 \) term. If you don't see an \( x^2 \) term at all, it means its coefficient is zero.

🎯 Exam Tip: Always look directly at the term containing the specified variable power. If the term is absent, its coefficient is 0. If it's just \( x^2 \), the coefficient is 1 (as \( 1 \cdot x^2 = x^2 \)).

Question 3. Given an example of a binomial of degree 45.
Answer: A binomial is a polynomial with exactly two terms. For it to have a degree of 45, the highest power of the variable in one of those terms must be 45. So, an example could be \( ax^{45} + b \), where 'a' is a non-zero real number (the coefficient of \( x^{45} \)) and 'b' is any non-zero constant. For instance, \( 5x^{45} + 3 \) is a binomial of degree 45. This demonstrates how a specific degree and number of terms define the polynomial.
In simple words: A binomial means it has two parts. Degree 45 means the biggest power on the letter is 45. So, an example is \( 5x^{45} + 3 \).

🎯 Exam Tip: For "binomial," remember "bi" means two terms. For "degree 45," ensure the highest power of the variable is exactly 45. Combine these to form your example.

Question 4. Give an example of a monomial of degree 120.
Answer: A monomial is a polynomial with exactly one term. For it to have a degree of 120, the power of the variable in that single term must be 120. So, an example could be \( ax^{120} \), where 'a' is a non-zero real number (the coefficient of \( x^{120} \)). For example, \( 7x^{120} \) is a monomial of degree 120. This simple form makes it easy to identify its degree and type.
In simple words: A monomial has only one part. Degree 120 means the power on the letter is 120. So, an example is \( 7x^{120} \).

🎯 Exam Tip: For "monomial," think "mono" means one term. For "degree 120," simply put the variable to the power of 120 and add a non-zero number in front of it.

Question 5. Give an example of a trinomial of degree 8.
Answer: A trinomial is a polynomial with exactly three terms. For it to have a degree of 8, the highest power of the variable in one of those terms must be 8. So, an example could be \( 2x^8 + 3x^4 + 5x \). This polynomial has three terms, and the highest power of x is 8, fitting the criteria for a trinomial of degree 8. You can use different coefficients and powers for the other terms, as long as the highest is 8.
In simple words: A trinomial has three parts. Degree 8 means the biggest power on the letter is 8. So, an example is \( 2x^8 + 3x^4 + 5x \).

🎯 Exam Tip: For "trinomial," remember "tri" means three terms. For "degree 8," ensure one term has \( x^8 \) and that 8 is the highest power. The other two terms can have lower powers.

Question 6. Can you write another terms in the example given question no. 3, 4, 5. If yes give two examples of each.
Answer: Yes, it is possible to write other examples for questions 3, 4, and 5 while still meeting the conditions. This shows the flexibility in constructing polynomials.
(i) For a binomial of degree 45 (like in Question 3), two other examples are: \( 2x^{45} + 3 \) and \( \frac {11}{2}x^{45} + 7 \).
(ii) For a monomial of degree 120 (like in Question 4), two other examples are: \( x^{120} \) and \( 7x^{120} \).
(iii) For a trinomial of degree 8 (like in Question 5), two other examples are: \( 7x^8 + 7x^5 + 1 \) and \( 17x^8 + 2x^5 + x \).
In simple words: Yes, you can make many different examples for these types of polynomials. You just need to keep the number of terms and the highest power the same.

🎯 Exam Tip: When asked for "other examples," remember you can change the coefficients (the numbers in front of the variables) and the powers of the other terms, as long as the definition (e.g., binomial, degree) remains consistent.

Question 7. Write the degree of each of the following polynomials.
(i) \( 12 - 3x + 2x^2 \)
(ii) \( 5y - \sqrt{2} \)
(iii) 9
(iv) \( 3 + 4t^2 \)
Answer:
(i) In the polynomial \( 12 - 3x + 2x^2 \), the powers of the variable x are 0 (for 12), 1 (for \( -3x \)), and 2 (for \( 2x^2 \)). The highest power among these is 2. So, its degree is 2.
(ii) In the polynomial \( 5y - \sqrt{2} \), the powers of the variable y are 1 (for \( 5y \)) and 0 (for \( -\sqrt{2} \)). The highest power of the variable y is 1. So, its degree is 1.
(iii) 9 is a constant polynomial. Constant polynomials, which are just numbers without variables, always have a degree of 0. For example, \( 9 = 9x^0 \).
(iv) In the polynomial \( 3 + 4t^2 \), the powers of the variable t are 0 (for 3) and 2 (for \( 4t^2 \)). The highest power of the variable t is 2. So, its degree is 2.
In simple words: The degree of a polynomial is the biggest power you see on any variable in the expression. If there are no variables, the degree is 0.

🎯 Exam Tip: To find the degree, identify all the variable powers in each term. The largest power found is the degree of the polynomial. For constant terms, the variable's power is considered to be 0.