GSEB Class 8 Maths Solutions Chapter 9 Algebraic Expressions and Identities Exercise 9.1

Question 1. Identify the terms, their coefficients for each of the following expressions?
(1) \( 5xyz^2 - 3zy \)
(2) \( 1 + x + x^2 \)
(3) \( 4x^2y^2 - 4x^2y^2z^2 + z^2 \)
(4) \( 3 - pq + qr - rp \)
(5) \( \frac{x}{2} + \frac{y}{2} - xy \)
(6) \( 0.3a - 0.6ab + 0.5b \)
Answer:
(1) The terms in \( 5xyz^2 - 3zy \) are \( 5xyz^2 \) and \( -3zy \). Their coefficients are \( 5 \) and \( -3 \) respectively.

(2) The terms in \( 1 + x + x^2 \) are \( 1 \), \( x \), and \( x^2 \). Their coefficients are \( 1 \), \( 1 \), and \( 1 \) respectively.

(3) The terms in \( 4x^2y^2 - 4x^2y^2z^2 + z^2 \) are \( 4x^2y^2 \), \( -4x^2y^2z^2 \), and \( z^2 \). Their coefficients are \( 4 \), \( -4 \), and \( 1 \) respectively.

(4) The terms in \( 3 - pq + qr - rp \) are \( 3 \), \( -pq \), \( qr \), and \( -rp \). Their coefficients are \( 3 \), \( -1 \), \( 1 \), and \( -1 \) respectively.

(5) The terms in \( \frac{x}{2} + \frac{y}{2} - xy \) are \( \frac{x}{2} \), \( \frac{y}{2} \), and \( -xy \). Their coefficients are \( \frac{1}{2} \), \( \frac{1}{2} \), and \( -1 \) respectively.

(6) The terms in \( 0.3a - 0.6ab + 0.5b \) are \( 0.3a \), \( -0.6ab \), and \( 0.5b \). Their coefficients are \( 0.3 \), \( -0.6 \), and \( 0.5 \) respectively.
In simple words: Each part of an expression separated by a plus or minus sign is called a term. The number that comes before the letters in a term is known as its coefficient. If there is no number, the coefficient is 1.

Exam Tip: Remember to include the sign (positive or negative) with the coefficient. A term like \( x \) has a coefficient of \( 1 \), and \( -y \) has a coefficient of \( -1 \).

 

Question 2. Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any of these three categories?
\( x + y \), \( 1000 \), \( x + x^2 + x^3 + x^4 \), \( 7 + y + 5x \), \( 2y - 3y^2 \), \( 2y - 3y^2 \), \( 2y - 3y^2 + 4y^3 \), \( 5x - 4y + 3xy \), \( 4z - 15z^2 \), \( ab + bc + cd + da \), \( pqr \), \( p^2q + pq^2 \), \( 2p + 2q \)
Answer: The polynomials are classified below based on the number of terms they contain.

The following polynomials do not fit into any of these categories:
* \( x + x^2 + x^3 + x^4 \) [It has 4 terms]
* \( ab + bc + cd + da \) [It also has 4 terms]
In simple words: Polynomials are named based on how many terms they have. One term is a monomial, two terms are a binomial, and three terms are a trinomial. If a polynomial has more than three terms, it doesn't fit these specific names.

Exam Tip: Always count the terms carefully to classify polynomials. Terms are separated by plus or minus signs. Make sure to identify like terms that might need to be combined first before counting.

 

Question 3. Add the following:
(i) \( ab - bc \), \( bc - ca \), \( ca - ab \)
(ii) \( a - b + ab \), \( b - c + bc \), \( c - a + ca \)
(iii) \( 2p^2q^2 - 3pq + 4 \), \( 5 + 7pq - 3p^2q^2 \)
(iv) \( l^2 + m^2 \), \( m^2 + n^2 \), \( n^2 + l^2 \), \( 2lm + 2mn + 2nl \)
Answer:
(i) We arrange the terms for vertical addition, grouping like terms together:

(ii) We arrange the terms vertically, making sure like terms align in columns:

(iii) We add the polynomials by grouping like terms:

(iv) We add all the given expressions by combining like terms:
In simple words: When you add polynomials, always put terms that are alike (meaning they have the same letters and powers) in columns. Then, add the numbers in front of those like terms. If a term doesn't have a matching partner, just bring it down as it is.

Exam Tip: For adding polynomials, arrange terms with the same variables and powers (like terms) in vertical columns. Add their coefficients while keeping the variables and powers unchanged. Make sure to include the signs correctly.

 

Question 4.
(a) Subtract \( 4a - 7ab + 3b + 12 \) from \( 12a - 9ab + 5b - 3 \)
(b) Subtract \( 3xy + 5yz - 7zx \) from \( 5xy - 2yz - 2zx + 10xyz \)
(c) Subtract \( 4p^2q - 3pq + 5pq^2 - 8p + 7q - 10 \) from \( 18 - 3p - 11q + 5pq - 2pq^2 + 5p^2q \)
Answer: For subtraction, we arrange the terms so that like terms are in the same column. Then, we change the sign of the terms being subtracted and add them.
(a) Subtract \( 4a - 7ab + 3b + 12 \) from \( 12a - 9ab + 5b - 3 \):

(b) Subtract \( 3xy + 5yz - 7zx \) from \( 5xy - 2yz - 2zx + 10xyz \):

(c) Subtract \( 4p^2q - 3pq + 5pq^2 - 8p + 7q - 10 \) from \( 18 - 3p - 11q + 5pq - 2pq^2 + 5p^2q \):
In simple words: To subtract polynomials, first arrange the terms so that the one being subtracted is below the other, with like terms lined up. Then, change the sign of every term in the bottom polynomial. Finally, add the terms vertically as you would normally.

Exam Tip: When subtracting polynomials, it is crucial to change the sign of every term in the polynomial being subtracted before combining like terms. A common mistake is forgetting to change all signs.