Maharashtra Board Class 9 Maths Chapter 3 Set 3.1 Algebra Standard Part 1 Polynomials Solutions

Question 1. State whether the given algebraic expressions are polynomials? Justify.
(i) \( y + \frac{1}{y} \)
(ii) \( 2 - 5\sqrt{x} \)
(iii) \( x^2 + 7x + 9 \)
(iv) \( 2m^{-2} + 7m - 5 \)
(v) 10
Answer:
(i) In the expression \( y + \frac{1}{y} = y + y^{-1} \), the index of the variable \( y \) in the term \( \frac{1}{y} \) is \( -1 \), which is not a whole number. Therefore, the given algebraic expression is not a polynomial.
(ii) In the expression \( 2 - 5\sqrt{x} = 2 - 5x^{\frac{1}{2}} \), the index of the variable \( x \) in the term \( 5\sqrt{x} \) is \( \frac{1}{2} \), which is not a whole number. Therefore, the given algebraic expression is not a polynomial.
(iii) In the expression \( x^2 + 7x + 9 \), the indices of the variable \( x \) are \( 2 \) and \( 1 \), which are whole numbers. Therefore, the given algebraic expression is a polynomial.
(iv) In the expression \( 2m^{-2} + 7m - 5 \), the index of the variable \( m \) in the term \( 2m^{-2} \) is \( -2 \), which is not a whole number. Therefore, the given algebraic expression is not a polynomial.
(v) 10 is a constant number which can be written as \( 10x^0 \). Here, the index of the variable is \( 0 \), which is a whole number. Therefore, 10 is a constant polynomial.
In simple words: An algebraic expression is called a polynomial only when all the powers of its variables are non-negative integers (whole numbers like 0, 1, 2, 3...). If any power is a fraction or a negative number, it cannot be a polynomial.

🎯 Exam Tip: To score full marks, always rewrite terms with denominators or roots into index form (like \( y^{-1} \) or \( x^{\frac{1}{2}} \)) before stating your justification.

Question 1. State whether the given algebraic expressions are polynomials. Justify.
(i) \( y + \frac{1}{y} \)
(ii) \( 2 - 5\sqrt{x} \)
(iii) \( x^2 + 7x + 9 \)
(iv) \( 2m^{-2} + 7m - 5 \)
(v) \( 10 \)
Answer:
(i) No, because the power of \( y \) in the term \( \frac{1}{y} \) is \( -1 \), which is a negative number and not a whole number.
(ii) No, because the power of \( x \) in the term \( 5\sqrt{x} \) is \( \frac{1}{2} \) (i.e., 0.5), which is a decimal and not a whole number.
(iii) Yes, because all the coefficients are real numbers and the power of each term is a whole number.
(iv) No, because the power of \( m \) in the term \( 2m^{-2} \) is \( -2 \), which is a negative number.
(v) Yes, because 10 is a constant polynomial with a degree of 0, which is a whole number.
In simple words: An expression is a polynomial only if the exponents of all its variables are non-negative whole numbers (like 0, 1, 2, 3...). If any variable is in the denominator or under a square root, it is not a polynomial.

🎯 Exam Tip: To quickly identify a polynomial, check if any variable is in the denominator or inside a radical sign; if so, the expression is not a polynomial.

Question 2. Write the coefficient of \( m^3 \) in each of the given polynomials.
(i) \( m^3 \)
(ii) \( -\frac{3}{2} + m - \sqrt{3}m^3 \)
(iii) \( -\frac{2}{3}m^3 + 5m^2 - 7m - 1 \)
Answer:
(i) 1
(ii) \( -\sqrt{3} \)
(iii) \( -\frac{2}{3} \). The coefficient is the real number multiplied by the variable term.
In simple words: The coefficient is simply the number written directly in front of the \( m^3 \) term. If there is no number shown, the coefficient is 1, and if there is a minus sign, make sure to include it.

🎯 Exam Tip: Always include the sign (positive or negative) that precedes the coefficient to avoid losing easy marks.

Question 3. Write the polynomial in \( x \) using the given information.
(i) Monomial with degree 7
(ii) Binomial with degree 35
(iii) Trinomial with degree 8
Answer:
(i) \( 5x^7 \) (or any single term with \( x^7 \))
(ii) \( x^{35} - 1 \) (or any two-term expression where the highest power is 35)
(iii) \( 3x^8 + 2x^6 + x^5 \) (or any three-term expression where the highest power is 8). Writing these expressions helps us understand how polynomials are structured based on their terms and degrees.
In simple words: A monomial has one term, a binomial has two terms, and a trinomial has three terms. The degree is the highest power of the variable in that expression.

🎯 Exam Tip: You can choose any real numbers as coefficients for these terms, as long as the number of terms and the highest power match the question's requirements.

Question 4. Write the degree of the given polynomials.
(i) \( \sqrt{5} \)
(ii) \( x^0 \)
(iii) \( x^2 \)
(iv) \( \sqrt{2}m^{10} - 7 \)
(v) \( 2p - \sqrt{7} \)
(vi) \( 7y - y^3 + y^5 \)
(vii) \( xyz + xy - z \)
Answer:
(i) 0 (since \( \sqrt{5} \) is a constant polynomial, which can be written as \( \sqrt{5}x^0 \))
(ii) 0
(iii) 2
(iv) 10
(v) 1 (since the power of \( p \) is 1)
(vi) 5 (since the highest power of \( y \) is 5)
(vii) 3 (the term \( xyz \) has three variables, each with a power of 1, so their sum is \( 1 + 1 + 1 = 3 \)). Finding the degree helps us classify polynomials and understand their behavior.
In simple words: The degree of a polynomial is the highest exponent of the variable. If a term has multiple variables multiplied together, you add their exponents to find the degree of that term.

🎯 Exam Tip: For terms with multiple variables like \( xyz \), remember to add the powers of all variables in that term to find its degree.

Question 4. Write the degree of each of the given polynomials.
(i) \(\sqrt{5}\)
(ii) \(x^0\)
(iii) \(x^2\)
(iv) \(\sqrt{2}m^{10} - 7\)
(v) \(2p - \sqrt{7}\)
(vi) \(7y - y^3 + y^5\)
(vii) \(xyz + xy - z\)
(viii) \(m^3n^7 - 3m^5n + mn\)
Answer:
(i) \(\sqrt{5} = \sqrt{5}x^0\)
\( \implies \) Degree of the polynomial = 0
(ii) \(x^0\)
\( \implies \) Degree of the polynomial = 0
(iii) \(x^2\)
\( \implies \) Degree of the polynomial = 2
(iv) \(\sqrt{2}m^{10} - 7\)
Here, the highest power of \(m\) is 10.
\( \implies \) Degree of the polynomial = 10
(v) \(2p - \sqrt{7}\)
Here, the highest power of \(p\) is 1.
\( \implies \) Degree of the polynomial = 1
(vi) \(7y - y^3 + y^5\)
Here, the highest power of \(y\) is 5.
\( \implies \) Degree of the polynomial = 5
(vii) \(xyz + xy - z\)
Here, the sum of the powers of \(x\), \(y\) and \(z\) in the term \(xyz\) is \(1 + 1 + 1 = 3\), which is the highest sum of powers in the given polynomial.
\( \implies \) Degree of the polynomial = 3
(viii) \(m^3n^7 - 3m^5n + mn\)
Here, the sum of the powers of \(m\) and \(n\) in the term \(m^3n^7\) is \(3 + 7 = 10\), which is the highest sum of powers in the given polynomial.
\( \implies \) Degree of the polynomial = 10
In simple words: The degree of a polynomial is the highest power of its variable. For terms with more than one variable, we add the powers of the variables in each term and take the highest sum.

🎯 Exam Tip: Remember that a constant number like \(\sqrt{5}\) has a degree of 0 because it can be written as \(\sqrt{5}x^0\). Always sum the powers of all variables in a single term when dealing with multi-variable polynomials.

Question 5. Classify the following polynomials as linear, quadratic and cubic polynomial.
(i) \(2x^2 + 3x + 1\)
(ii) \(5p\)
Answer:
(i) \(2x^2 + 3x + 1\)
The degree of this polynomial is 2. Therefore, it is a quadratic polynomial.
(ii) \(5p\)
The degree of this polynomial is 1. Therefore, it is a linear polynomial.
In simple words: A polynomial with degree 1 is linear, degree 2 is quadratic, and degree 3 is cubic.

🎯 Exam Tip: To classify a polynomial, first find its highest power (degree). If the highest power is 1, it is linear; if 2, it is quadratic; if 3, it is cubic.

Question 6. Write the following polynomials in standard form.
(i) \( m^3 + 3 + 5m \)
(ii) \( -7y + y^5 + 3y^3 - \frac{1}{2} + 2y^4 - y^2 \)
Answer:
(i) \( m^3 + 5m + 3 \)
(ii) \( y^5 + 2y^4 + 3y^3 - y^2 - 7y - \frac{1}{2} \)
In simple words: Writing a polynomial in standard form means arranging its terms from the highest power of the variable down to the lowest power.

🎯 Exam Tip: Always identify the term with the highest exponent first and arrange the remaining terms in descending order of their powers, keeping their original positive or negative signs intact.

Question 7. Write the following polynomials in coefficient form.
(i) \( x^3 - 2 \)
(ii) \( 5y \)
(iii) \( 2m^4 - 3m^2 + 7 \)
(iv) \( -\frac{2}{3} \)
Answer:
(i) \( x^3 - 2 = x^3 + 0x^2 + 0x - 2 \)
\( \implies \) Coefficient form of the given polynomial = \( (1, 0, 0, -2) \)
(ii) \( 5y = 5y + 0 \)
\( \implies \) Coefficient form of the given polynomial = \( (5, 0) \)
(iii) \( 2m^4 - 3m^2 + 7 = 2m^4 + 0m^3 - 3m^2 + 0m + 7 \)
\( \implies \) Coefficient form of the given polynomial = \( (2, 0, -3, 0, 7) \)
(iv) \( -\frac{2}{3} \)
\( \implies \) Coefficient form of the given polynomial = \( \left(-\frac{2}{3}\right) \)
In simple words: To write a polynomial in coefficient form, first fill in any missing powers of the variable with a coefficient of 0, and then list only the numerical coefficients inside parentheses.

🎯 Exam Tip: A common mistake is forgetting to write 0 for the missing index terms. Always write the polynomial in its complete index form before extracting the coefficients.

Question 8. Write the polynomials in index form.
(i) \( (1, 2, 3) \)
(ii) \( (5, 0, 0, 0, -1) \)
(iii) \( (-2, 2, -2, 2) \)
Answer:
(i) Number of coefficients = \( 3 \)
\( \therefore \) Degree = \( 3 - 1 = 2 \)
\( \dots \) Taking \( x \) as variable, the index form is \( x^2 + 2x + 3 \). This represents a standard quadratic expression.
(ii) Number of coefficients = \( 5 \)
\( \therefore \) Degree = \( 5 - 1 = 4 \)
\( \dots \) Taking \( x \) as variable, the index form is \( 5x^4 + 0x^3 + 0x^2 + 0x - 1 \). Writing all terms helps us keep track of the place values of each power.
(iii) Number of coefficients = \( 4 \)
\( \therefore \) Degree = \( 4 - 1 = 3 \)
\( \dots \) Taking \( x \) as variable, the index form is \( -2x^3 + 2x^2 - 2x + 2 \).
In simple words: To write a polynomial from its coefficients, count how many numbers there are and subtract 1 to find the highest power (degree). Then, write the powers of x from highest to lowest, multiplying each by its corresponding coefficient.

🎯 Exam Tip: Always remember that the degree of the polynomial is always one less than the total number of coefficients given in the bracket.

Question 9. Write the appropriate polynomials in the boxes.
Given polynomials in the diagram:

• \( x + 7 \)
• \( x^2 \)
• \( x^3 + x^2 + x + 5 \)
• \( 2x^2 + 5x + 10 \)
• \( x^3 + 9 \)
• \( 3x^2 + 5x \)

Answer:
(i) Quadratic polynomial: \( x^2 \); \( 2x^2 + 5x + 10 \); \( 3x^2 + 5x \)
(ii) Cubic polynomial: \( x^3 + x^2 + x + 5 \); \( x^3 + 9 \)
(iii) Linear polynomial: \( x + 7 \)
(iv) Binomial: \( x + 7 \); \( x^3 + 9 \); \( 3x^2 + 5x \)
(v) Trinomial: \( 2x^2 + 5x + 10 \)
(vi) Monomial: \( x^2 \). Classifying these expressions helps us understand their structure and behavior in algebraic operations.
In simple words: Polynomials are classified in two ways: by their highest power (linear, quadratic, cubic) and by how many terms they have (monomial has one term, binomial has two, trinomial has three).

🎯 Exam Tip: Be careful not to confuse classification by degree (linear, quadratic, cubic) with classification by number of terms (monomial, binomial, trinomial).

Question 1. Write an example of a monomial, a binomial and a trinomial having variable x and degree 5. (Textbook pg. no. 3)
Answer:
Monomial: \( x^5 \)
Binomial: \( x^5 + x \)
Trinomial: \( 2x^5 - x^2 + 5 \). These examples clearly demonstrate how polynomials are classified based on the number of terms they contain.
In simple words: A monomial has only one term, a binomial has two terms, and a trinomial has three terms. The degree is the highest power of the variable, which is 5 in each of these examples.

🎯 Exam Tip: Remember that the degree of a polynomial is the highest exponent of the variable. Make sure your examples have exactly 1, 2, and 3 terms respectively with the highest power as 5.

Question 2. Give example of a binomial in two variables having degree 5. (Textbook pg. no. 38)
Answer: \( x^3y^2 + xy \). This expression uses two different variables, x and y, where the sum of the powers in the highest term equals 5.
In simple words: A binomial must have exactly two terms. To find the degree when there are two variables in a term, we add their powers together, so 3 + 2 gives us a degree of 5.

🎯 Exam Tip: When finding the degree of a term with multiple variables, always add the exponents of all the variables in that term together.