Samacheer Kalvi Class 9 Maths Solutions Chapter 3 Algebra Exercise 3.1

Question 1. Which of the following expressions are polynomials. If not give reason:
(i) \( \frac{1}{x^2} + 3x-4 \)
(ii) \( x^2 (x – 1) \)
(iii) \( \frac{1}{x} (x + 5) \)
(iv) \( \frac{1}{x^{-2}} + \frac{1}{x^{-1}} + 7 \)
(v) \( \sqrt{5}x^2 + \sqrt{3}x + \sqrt{2} \)
(vi) \( m^2 - \sqrt[3]{m} + 7m – 10 \)
Answer:
(i) The expression \( \frac{1}{x^2} + 3x - 4 \) is not a polynomial. This is because the term \( \frac{1}{x^2} \) can be rewritten as \( x^{-2} \), and the exponent \( -2 \) is a negative number. For an expression to be a polynomial, all exponents of the variables must be non-negative whole numbers.
(ii) The expression \( x^2 (x - 1) \) is a polynomial. When we multiply it out, it becomes \( x^3 - x^2 \). All the exponents ( \( 3 \) and \( 2 \) ) are positive whole numbers.
(iii) The expression \( \frac{1}{x} (x + 5) \) is not a polynomial. This is because the term \( \frac{1}{x} \) is the same as \( x^{-1} \), and the exponent \( -1 \) is a negative number. Polynomials must have only whole number exponents for their variables.
(iv) The expression \( \frac{1}{x^{-2}} + \frac{1}{x^{-1}} + 7 \) is a polynomial. We can simplify \( \frac{1}{x^{-2}} \) to \( x^2 \) and \( \frac{1}{x^{-1}} \) to \( x \). So, the expression becomes \( x^2 + x + 7 \), where all exponents ( \( 2 \) and \( 1 \) ) are positive whole numbers.
(v) The expression \( \sqrt{5}x^2 + \sqrt{3}x + \sqrt{2} \) is a polynomial. Even though the numbers multiplying \( x^2 \) and \( x \) (the coefficients) are square roots, the exponents of the variable \( x \) are \( 2 \) and \( 1 \), which are positive whole numbers.
(vi) The expression \( m^2 - \sqrt[3]{m} + 7m - 10 \) is not a polynomial. This is because the term \( \sqrt[3]{m} \) can be written as \( m^{1/3} \), and the exponent \( \frac{1}{3} \) is a fraction, not a whole number.
In simple words: A polynomial has variables with only positive whole number powers (like \( x^2, x^3 \)). If you see negative powers (like \( x^{-2} \)) or fractional powers (like \( \sqrt{x} \) or \( x^{1/3} \)), it's not a polynomial.

🎯 Exam Tip: Remember that coefficients can be any real number (like \( \sqrt{5} \) or fractions), but the exponents of the variables must always be non-negative integers for an expression to be a polynomial.

Question 2. Write the coefficient of \( x^2 \) and \( x \) in each of the following polynomials.
(i) \( 4 + \frac{2}{5} x^2 – 3x \)
(ii) \( 6 - 2x^2 + 3x^3 – \sqrt{7}x \)
(iii) \( \pi x^2 - x + 2 \)
(iv) \( \sqrt{3}x^2 + \sqrt{2}x + 0.5 \)
(v) \( x^2 - \frac{7}{2} x + 8 \)
Answer:
(i) In the polynomial \( 4 + \frac{2}{5} x^2 - 3x \):
The coefficient of \( x^2 \) is \( \frac{2}{5} \).
The coefficient of \( x \) is \( -3 \).
(ii) In the polynomial \( 6 - 2x^2 + 3x^3 - \sqrt{7}x \):
The coefficient of \( x^2 \) is \( -2 \).
The coefficient of \( x \) is \( -\sqrt{7} \).
(iii) In the polynomial \( \pi x^2 - x + 2 \):
The coefficient of \( x^2 \) is \( \pi \).
The coefficient of \( x \) is \( -1 \) (since \( -x \) means \( -1 \times x \)).
(iv) In the polynomial \( \sqrt{3}x^2 + \sqrt{2}x + 0.5 \):
The coefficient of \( x^2 \) is \( \sqrt{3} \).
The coefficient of \( x \) is \( \sqrt{2} \).
(v) In the polynomial \( x^2 - \frac{7}{2} x + 8 \):
The coefficient of \( x^2 \) is \( 1 \) (since \( x^2 \) means \( 1 \times x^2 \)).
The coefficient of \( x \) is \( -\frac{7}{2} \).
In simple words: The coefficient is the number that is multiplied by the variable in a term. Remember to include its sign. If a variable has no number in front, its coefficient is 1.

🎯 Exam Tip: Always pay close attention to the sign (positive or negative) in front of each term when identifying coefficients, as a missing sign means it's positive.

Question 3. Find the degree of the following polynomials.
(i) \( 1 - \sqrt{2} y^2 + y^7 \)
(ii) \( \frac{x^{3}-x^{4}+6 x^{6}}{x^{2}} \)
(iii) \( x^3 (x^2 + x) \)
(iv) \( 3x^4 + 9x^2 + 27x^6 \)
(v) \( 2\sqrt{5}p^4 - \frac{8 p^{3}}{\sqrt{3}} + \frac{2 p^{2}}{7} \)
Answer:
(i) In the polynomial \( 1 - \sqrt{2} y^2 + y^7 \), the highest power (exponent) of the variable \( y \) is \( 7 \). Therefore, the degree of this polynomial is \( 7 \).
(ii) First, we simplify the expression \( \frac{x^{3}-x^{4}+6 x^{6}}{x^{2}} \). Divide each term in the numerator by \( x^2 \):
\( \frac{x^3}{x^2} - \frac{x^4}{x^2} + \frac{6x^6}{x^2} = x - x^2 + 6x^4 \)
In the simplified polynomial \( x - x^2 + 6x^4 \), the highest power of \( x \) is \( 4 \). So, the degree of the polynomial is \( 4 \).
(iii) First, we multiply the terms in \( x^3 (x^2 + x) \):
\( x^3 \times x^2 + x^3 \times x = x^{3+2} + x^{3+1} = x^5 + x^4 \)
In the polynomial \( x^5 + x^4 \), the highest power of \( x \) is \( 5 \). Therefore, the degree of the polynomial is \( 5 \).
(iv) In the polynomial \( 3x^4 + 9x^2 + 27x^6 \), the exponents of \( x \) are \( 4, 2, \) and \( 6 \). The largest among these is \( 6 \). So, the degree of the polynomial is \( 6 \).
(v) In the polynomial \( 2\sqrt{5}p^4 - \frac{8 p^{3}}{\sqrt{3}} + \frac{2 p^{2}}{7} \), the exponents of the variable \( p \) are \( 4, 3, \) and \( 2 \). The highest exponent is \( 4 \). Therefore, the degree of the polynomial is \( 4 \).
In simple words: The degree of a polynomial is simply the biggest power you see on any variable in the expression. If you need to multiply or divide terms first, do that to get the simplest form, then find the highest power.

🎯 Exam Tip: Always remember to simplify the polynomial completely (expand, combine like terms, or perform division) before identifying its highest exponent to determine the correct degree.

Question 4. Rewrite the following polynomial in standard form.
(i) \( x - 9 + \sqrt{7}x^3 + 6x^2 \)
(ii) \( \sqrt{2}x^2 - \frac{7}{2} x^4 + x - 5x^3 \)
(iii) \( 7x^3 - \frac{6}{5} x^2 + 4x - 1 \)
(iv) \( y^2 + \sqrt{5}y^3 - 11 - \frac{7}{3} y + 9y^4 \)
Answer:
(i) To write \( x - 9 + \sqrt{7}x^3 + 6x^2 \) in standard form, we arrange the terms in descending order of their exponents (highest to lowest power of \( x \)).
The terms are \( \sqrt{7}x^3 \) (degree 3), \( 6x^2 \) (degree 2), \( x \) (degree 1), and \( -9 \) (degree 0, constant term).
So, the standard form is \( \sqrt{7}x^3 + 6x^2 - x - 9 \).
(ii) To write \( \sqrt{2}x^2 - \frac{7}{2} x^4 + x - 5x^3 \) in standard form, arrange the terms by their powers from highest to lowest.
The terms are \( -\frac{7}{2} x^4 \) (degree 4), \( -5x^3 \) (degree 3), \( \sqrt{2}x^2 \) (degree 2), and \( x \) (degree 1).
So, the standard form is \( -\frac{7}{2} x^4 - 5x^3 + \sqrt{2}x^2 + x \).
(iii) To write \( 7x^3 - \frac{6}{5} x^2 + 4x - 1 \) in standard form, arrange the terms from the highest power of \( x \) to the lowest.
The terms are \( 7x^3 \) (degree 3), \( -\frac{6}{5} x^2 \) (degree 2), \( 4x \) (degree 1), and \( -1 \) (degree 0).
This polynomial is already in standard form: \( 7x^3 - \frac{6}{5} x^2 + 4x - 1 \).
(iv) To write \( y^2 + \sqrt{5}y^3 - 11 - \frac{7}{3} y + 9y^4 \) in standard form, arrange the terms by their powers from highest to lowest.
The terms are \( 9y^4 \) (degree 4), \( \sqrt{5}y^3 \) (degree 3), \( y^2 \) (degree 2), \( -\frac{7}{3} y \) (degree 1), and \( -11 \) (degree 0).
So, the standard form is \( 9y^4 + \sqrt{5}y^3 + y^2 - \frac{7}{3} y - 11 \).
In simple words: Standard form means writing the polynomial terms in order, starting with the highest power of the variable and going down to the lowest. The number without a variable always comes last.

🎯 Exam Tip: Always include the sign of each term when rearranging for standard form, and ensure the constant term (if any) is placed at the very end.

Question 5. Add the following polynomials and find the degree of the resultant polynomial
(i) \( p(x) = 6x^2 – 7x + 2; q(x) = 6x^3 – 7x + 15 \)
(ii) \( h(x) = 7x^3 – 6x + 1; f(x) = 7x^2 + 17x – 9 \)
(iii) \( f(x) = 16x^4 – 5x^2 + 9; g(x) = -6x^3 + 7x – 15 \)
Answer:
(i) To add \( p(x) \) and \( q(x) \):
\( p(x) + q(x) = (6x^2 - 7x + 2) + (6x^3 - 7x + 15) \)
Group like terms:
\( = 6x^3 + 6x^2 + (-7x - 7x) + (2 + 15) \)
Combine like terms:
\( = 6x^3 + 6x^2 - 14x + 17 \)
The highest power of \( x \) in the resultant polynomial is \( 3 \). So, the degree is \( 3 \).
(ii) To add \( h(x) \) and \( f(x) \):
\( h(x) + f(x) = (7x^3 - 6x + 1) + (7x^2 + 17x - 9) \)
Group like terms:
\( = 7x^3 + 7x^2 + (-6x + 17x) + (1 - 9) \)
Combine like terms:
\( = 7x^3 + 7x^2 + 11x - 8 \)
The highest power of \( x \) in the resultant polynomial is \( 3 \). So, the degree is \( 3 \).
(iii) To add \( f(x) \) and \( g(x) \):
\( f(x) + g(x) = (16x^4 - 5x^2 + 9) + (-6x^3 + 7x - 15) \)
Group like terms:
\( = 16x^4 - 6x^3 - 5x^2 + 7x + (9 - 15) \)
Combine like terms:
\( = 16x^4 - 6x^3 - 5x^2 + 7x - 6 \)
The highest power of \( x \) in the resultant polynomial is \( 4 \). So, the degree is \( 4 \).
In simple words: To add polynomials, just put terms with the same variable and power together and add their numbers. The degree of the new polynomial will be the highest power in the final answer.

🎯 Exam Tip: When adding polynomials, organize terms vertically or horizontally by degree to reduce errors in combining like terms and accurately identifying the resultant degree.

Question 6. Subtract the second polynomial from the first polynomial and find the degree of the resultant polynomial.
(i) \( p(x) = 7x^2 + 6x – 1; q(x) = 6x – 9 \)
(ii) \( f(y) = 6y^2 – 7y + 2; g(y) = 7y + y^3 \)
(iii) \( h(z) = z^5 – 6z^4 + z; f(z) = 6z^2 + 10z – 7 \)
Answer:
(i) To subtract \( q(x) \) from \( p(x) \):
\( p(x) - q(x) = (7x^2 + 6x - 1) - (6x - 9) \)
Distribute the negative sign:
\( = 7x^2 + 6x - 1 - 6x + 9 \)
Group and combine like terms:
\( = 7x^2 + (6x - 6x) + (-1 + 9) \)
\( = 7x^2 + 0 + 8 \)
\( = 7x^2 + 8 \)
The highest power of \( x \) in the resultant polynomial is \( 2 \). So, the degree is \( 2 \).
(ii) To subtract \( g(y) \) from \( f(y) \):
\( f(y) - g(y) = (6y^2 - 7y + 2) - (7y + y^3) \)
Distribute the negative sign:
\( = 6y^2 - 7y + 2 - 7y - y^3 \)
Group and combine like terms, arranging in descending order of powers:
\( = -y^3 + 6y^2 + (-7y - 7y) + 2 \)
\( = -y^3 + 6y^2 - 14y + 2 \)
The highest power of \( y \) in the resultant polynomial is \( 3 \). So, the degree is \( 3 \).
(iii) To subtract \( f(z) \) from \( h(z) \):
\( h(z) - f(z) = (z^5 - 6z^4 + z) - (6z^2 + 10z - 7) \)
Distribute the negative sign:
\( = z^5 - 6z^4 + z - 6z^2 - 10z + 7 \)
Group and combine like terms, arranging in descending order of powers:
\( = z^5 - 6z^4 - 6z^2 + (z - 10z) + 7 \)
\( = z^5 - 6z^4 - 6z^2 - 9z + 7 \)
The highest power of \( z \) in the resultant polynomial is \( 5 \). So, the degree is \( 5 \).
In simple words: When subtracting polynomials, change the sign of every term in the second polynomial, then combine like terms. The degree of the final polynomial is its highest power.

🎯 Exam Tip: The most common mistake in polynomial subtraction is forgetting to distribute the negative sign to *all* terms of the polynomial being subtracted.

Question 7. What should be added to \( 2x^3 + 6x^2 – 5x + 8 \) to get \( 3x^3 – 2x^2 + 6x + 15 \)?
Answer:
Let the unknown polynomial be \( P \). We want to find \( P \) such that:
\( (2x^3 + 6x^2 - 5x + 8) + P = (3x^3 - 2x^2 + 6x + 15) \)
To find \( P \), subtract the first polynomial from the second:
\( P = (3x^3 - 2x^2 + 6x + 15) - (2x^3 + 6x^2 - 5x + 8) \)
Distribute the negative sign:
\( P = 3x^3 - 2x^2 + 6x + 15 - 2x^3 - 6x^2 + 5x - 8 \)
Group and combine like terms:
\( P = (3x^3 - 2x^3) + (-2x^2 - 6x^2) + (6x + 5x) + (15 - 8) \)
\( P = x^3 - 8x^2 + 11x + 7 \)
So, \( x^3 - 8x^2 + 11x + 7 \) must be added to get the desired polynomial.
In simple words: To find out what you need to add, you take the target polynomial and subtract the starting polynomial from it. This is like asking "what do I add to 5 to get 10?" The answer is \( 10 - 5 = 5 \).

🎯 Exam Tip: When setting up this type of problem, clearly define which polynomial is being subtracted from which to avoid calculation errors.

Question 8. What must be subtracted from \( 2x^4 + 4x^2 – 3x + 7 \) to get \( 3x^3 – x^2 + 2x + 1 \)?
Answer:
Let the unknown polynomial be \( S \). We want to find \( S \) such that:
\( (2x^4 + 4x^2 - 3x + 7) - S = (3x^3 - x^2 + 2x + 1) \)
To find \( S \), rearrange the equation:
\( S = (2x^4 + 4x^2 - 3x + 7) - (3x^3 - x^2 + 2x + 1) \)
Distribute the negative sign:
\( S = 2x^4 + 4x^2 - 3x + 7 - 3x^3 + x^2 - 2x - 1 \)
Group and combine like terms, arranging in descending order of powers:
\( S = 2x^4 - 3x^3 + (4x^2 + x^2) + (-3x - 2x) + (7 - 1) \)
\( S = 2x^4 - 3x^3 + 5x^2 - 5x + 6 \)
So, \( 2x^4 - 3x^3 + 5x^2 - 5x + 6 \) must be subtracted.
In simple words: To find what polynomial to subtract, you take the first polynomial and subtract the desired result from it. Think of it as "what do I subtract from 10 to get 3?" The answer is \( 10 - 3 = 7 \).

🎯 Exam Tip: For "what must be subtracted" questions, always subtract the target result from the starting polynomial. This ensures the correct unknown polynomial is found.

Question 9. Multiply the following polynomials and find the degree of the resultant polynomial:
(i) \( p(x) = x^2 – 9, q(x) = 6x^2 + 7x – 2 \)
(ii) \( f(x) = 7x + 2, g(x) = 15x –9 \)
(iii) \( h(x) = 6x^2 – 7x + 1, f(x) = 5x – 7 \)
Answer:
(i) To multiply \( p(x) \) and \( q(x) \):
\( p(x) \times q(x) = (x^2 - 9)(6x^2 + 7x - 2) \)
Distribute each term from the first polynomial to the second:
\( = x^2(6x^2 + 7x - 2) - 9(6x^2 + 7x - 2) \)
\( = 6x^4 + 7x^3 - 2x^2 - 54x^2 - 63x + 18 \)
Combine like terms:
\( = 6x^4 + 7x^3 + (-2x^2 - 54x^2) - 63x + 18 \)
\( = 6x^4 + 7x^3 - 56x^2 - 63x + 18 \)
The highest power of \( x \) in the resultant polynomial is \( 4 \). So, the degree is \( 4 \).
(ii) To multiply \( f(x) \) and \( g(x) \):
\( f(x) \times g(x) = (7x + 2)(15x - 9) \)
Using the FOIL method (First, Outer, Inner, Last):
\( = (7x)(15x) + (7x)(-9) + (2)(15x) + (2)(-9) \)
\( = 105x^2 - 63x + 30x - 18 \)
Combine like terms:
\( = 105x^2 + (-63x + 30x) - 18 \)
\( = 105x^2 - 33x - 18 \)
The highest power of \( x \) in the resultant polynomial is \( 2 \). So, the degree is \( 2 \).
(iii) To multiply \( h(x) \) and \( f(x) \):
\( h(x) \times f(x) = (6x^2 - 7x + 1)(5x - 7) \)
Distribute each term from the first polynomial to the second:
\( = 6x^2(5x - 7) - 7x(5x - 7) + 1(5x - 7) \)
\( = (30x^3 - 42x^2) - (35x^2 - 49x) + (5x - 7) \)
\( = 30x^3 - 42x^2 - 35x^2 + 49x + 5x - 7 \)
Combine like terms:
\( = 30x^3 + (-42x^2 - 35x^2) + (49x + 5x) - 7 \)
\( = 30x^3 - 77x^2 + 54x - 7 \)
The highest power of \( x \) in the resultant polynomial is \( 3 \). So, the degree is \( 3 \).
In simple words: To multiply polynomials, make sure every term from the first polynomial is multiplied by every term from the second. Then, add together all the terms that have the same variable and power. The degree of the final answer will be the sum of the degrees of the original polynomials.

🎯 Exam Tip: Be methodical when multiplying polynomials; ensure every term is multiplied by every other term, and be careful with signs and combining like terms. The degree of the product is the sum of the degrees of the factors.

Question 10. The cost of a chocolate is Rs. \( (x + y) \) and Amir bought \( (x + y) \) chocolates. Find the total amount paid by him in terms of \( x \) and \( y \). If \( x = 10, y = 5 \) find the amount paid by him.
Answer:
The cost of one chocolate is Rs. \( (x+y) \).
The number of chocolates bought by Amir is \( (x+y) \).
Total amount paid = Cost of one chocolate \( \times \) Number of chocolates
Total amount paid \( = (x+y) \times (x+y) \)
\( = (x+y)^2 \)
Using the identity \( (a+b)^2 = a^2 + 2ab + b^2 \):
\( = x^2 + 2xy + y^2 \)
This is the total amount paid in terms of \( x \) and \( y \).

Now, substitute \( x = 10 \) and \( y = 5 \) into the expression:
Total amount paid \( = (10)^2 + 2(10)(5) + (5)^2 \)
\( = 100 + 100 + 25 \)
\( = 225 \)
Therefore, if \( x=10 \) and \( y=5 \), the total amount paid by Amir is Rs. \( 225 \).
In simple words: If something costs \( (x+y) \) and you buy \( (x+y) \) of them, you pay \( (x+y) \) multiplied by itself. That's \( (x+y)^2 \), which is \( x^2 + 2xy + y^2 \). Then, put in the numbers for \( x \) and \( y \) to get the final amount.

🎯 Exam Tip: For word problems involving algebraic expressions, first set up the general formula using variables, then substitute specific numerical values to find the final answer.

Question 11. The length of a rectangle is \( (3x + 2) \) units and it's breadth is \( (3x – 2) \) units. Find its area in terms of \( x \). What will be the area if \( x = 20 \) units.
Answer:
Given:
Length of the rectangle \( = (3x + 2) \) units
Breadth of the rectangle \( = (3x - 2) \) units

The area of a rectangle is Length \( \times \) Breadth.
Area \( = (3x + 2)(3x - 2) \)
This is in the form of the algebraic identity \( (a+b)(a-b) = a^2 - b^2 \).
Here, \( a = 3x \) and \( b = 2 \).
So, Area \( = (3x)^2 - (2)^2 \)
\( = 9x^2 - 4 \)
This is the area of the rectangle in terms of \( x \).

Now, find the area if \( x = 20 \) units.
Substitute \( x = 20 \) into the area expression:
Area \( = 9(20)^2 - 4 \)
\( = 9(400) - 4 \)
\( = 3600 - 4 \)
\( = 3596 \)
So, if \( x = 20 \) units, the area of the rectangle is \( 3596 \) square units.
In simple words: To find the area, multiply the length and breadth. Since they are in the form \( (A+B)(A-B) \), the area is \( A^2 - B^2 \). Then, put the given value of \( x \) into the area formula to find the number for the area.

🎯 Exam Tip: Recognizing and applying algebraic identities like the difference of squares (\( a^2 - b^2 \)) can greatly simplify multiplication of binomials, saving time and reducing error.

Question 12. \( p(x) \) is a polynomial of degree 1 and \( q(x) \) is a polynomial of degree 2. What kind of the polynomial is \( p(x) \times q(x) \)?
Answer:
Given:
Degree of polynomial \( p(x) = 1 \)
Degree of polynomial \( q(x) = 2 \)

When two polynomials are multiplied, the degree of the resultant polynomial is the sum of the degrees of the individual polynomials.
Degree of \( (p(x) \times q(x)) = \text{Degree of } p(x) + \text{Degree of } q(x) \)
\( = 1 + 2 \)
\( = 3 \)
A polynomial with a degree of \( 3 \) is called a cubic polynomial.
Therefore, \( p(x) \times q(x) \) is a cubic polynomial.
In simple words: When you multiply two polynomials, you simply add their highest powers together to find the highest power of the new polynomial. If the highest power is 3, it's called a cubic polynomial.

🎯 Exam Tip: Always remember that the degree of a product of polynomials is the sum of their individual degrees, while for addition/subtraction, it's generally the highest degree among them.