CBSE Class 8 Mathematics Algebraic Expressions And Identities Worksheet Set 03

Question 1. What must be subtracted from \(-9x^3y + 3xy^2 + 4x - 8y + 3\) to get \(9x^3y - 8xy^2 + 3x - 2y + 6\).
Answer: Let the required expression to be subtracted be \(P\).
According to the question:
\((-9x^3y + 3xy^2 + 4x - 8y + 3) - P = 9x^3y - 8xy^2 + 3x - 2y + 6\)
Rearranging for \(P\):
\(P = (-9x^3y + 3xy^2 + 4x - 8y + 3) - (9x^3y - 8xy^2 + 3x - 2y + 6)\)
\(P = -9x^3y + 3xy^2 + 4x - 8y + 3 - 9x^3y + 8xy^2 - 3x + 2y - 6\)
Grouping like terms together:
\(P = (-9x^3y - 9x^3y) + (3xy^2 + 8xy^2) + (4x - 3x) + (-8y + 2y) + (3 - 6)\)
\(P = -18x^3y + 11xy^2 + x - 6y - 3\)
In simple words: To find what must be subtracted, subtract the target expression from the given expression.

Exam Tip: Be very careful with signs when distributing the negative sign across the second polynomial.

Question 2. Arrange and add: \(3a^2b + 4ab^2 + 7a^2b^2 - 8ab\); \(8a^2b^2 + 2ab - 2ab^2\); \(5ab - 4a^2b + a^2b^2 - 6ab^2\).
Answer: First, write down all the expressions by grouping like terms in columns:
Expression 1: \(3a^2b + 4ab^2 + 7a^2b^2 - 8ab\)
Expression 2: \(-4a^2b - 6ab^2 + a^2b^2 + 5ab\) (rearranged by like terms)
Expression 3: \(-2ab^2 + 8a^2b^2 + 2ab\) (rearranged by like terms)
Now, add the corresponding coefficients column-wise:
For \(a^2b\) terms: \(3 - 4 = -1\), so \(-a^2b\)
For \(ab^2\) terms: \(4 - 6 - 2 = -4\), so \(-4ab^2\)
For \(a^2b^2\) terms: \(7 + 1 + 8 = 16\), so \(16a^2b^2\)
For \(ab\) terms: \(-8 + 5 + 2 = -1\), so \(-ab\)
Combining all terms: \(-a^2b - 4ab^2 + 16a^2b^2 - ab\)
In simple words: Align all terms with the exact same variables and powers, then add or subtract their numbers.

Exam Tip: Always rewrite expressions in standard alphabetical or degree order before adding columns to avoid missing any terms.

Question 3. Fill in the blanks:
a. \(-8x^2y \times \text{______} = 16x^3y^3\)
Answer: \(-2xy^2\)
Step-by-step calculation:
Let the missing term be \(M\).
\(M = \frac{16x^3y^3}{-8x^2y} = \left(\frac{16}{-8}\right) \cdot \left(\frac{x^3}{x^2}\right) \cdot \left(\frac{y^3}{y}\right) = -2x^{3-2}y^{3-1} = -2xy^2\)
In simple words: Divide the product by the given term using exponent laws.

Exam Tip: Use exponent rule \(\frac{x^m}{x^n} = x^{m-n}\) for dividing variables.

b. \(5p^2q^3r^2 \times \text{______} = 0\)
Answer: \(0\)
Step-by-step calculation:
Any term multiplied by \(0\) results in \(0\). Thus, the missing term must be \(0\).
In simple words: Multiplying any expression by zero always yields zero.

Exam Tip: Look out for zero products instantly to save time.

c. \(-2abc \times -5abc = \text{______}\)
Answer: \(10a^2b^2c^2\)
Step-by-step calculation:
\((-2) \times (-5) = 10\)
\(a \times a = a^2\)
\(b \times b = b^2\)
\(c \times c = c^2\)
Product = \(10a^2b^2c^2\)
In simple words: Multiply the numbers together and add the powers of matching variables.

Exam Tip: Minus times minus gives a plus sign.

Question 4. Multiply and fill in the blanks
a. \(5pq(p^2 + q^2 + pq) = \text{____________________}\)
Answer: \(5p^3q + 5pq^3 + 5p^2q^2\)
Step-by-step calculation:
\(5pq \times p^2 = 5p^3q\)
\(5pq \times q^2 = 5pq^3\)
\(5pq \times pq = 5p^2q^2\)
Combined: \(5p^3q + 5pq^3 + 5p^2q^2\)
In simple words: Multiply the monomial outside by each term inside the bracket one by one.

Exam Tip: Apply distributive property: \(a(b+c+d) = ab + ac + ad\).

b. \(-7pr(pq - qr - rq) = \text{_________________________}\)
Answer: \(-7p^2q जैसा + 7pqr^2 + 7pqr^2\) or \(-7p^2qr + 14pqr^2\)
Step-by-step calculation:
Note that \(-qr\) and \(-rq\) are like terms: \(-qr - rq = -2qr\).
\(-7pr(pq - 2qr) = (-7pr \times pq) - (-7pr \times 2qr)\)
\(= -7p^2qr + 14pqr^2\)
In simple words: Combine like terms inside the bracket first, then multiply.

Exam Tip: Always simplify inside parentheses before multiplying to avoid errors.

c. \((4x - 3y)(5x - 4a) = \text{_____________________________}\)
Answer: \(20x^2 - 16ax - 15xy + 12ay\)
Step-by-step calculation:
\(= 4x(5x - 4a) - 3y(5x - 4a)\)
\(= (4x \times 5x) + (4x \times -4a) + (-3y \times 5x) + (-3y \times -4a)\)
\(= 20x^2 - 16ax - 15xy + 12ay\)
In simple words: Multiply every term in the first binomial by every term in the second binomial (FOIL method).

Exam Tip: Remember the FOIL acronym: First, Outer, Inner, Last.

Question 5. The length and breadth of a rectangle are \((3a - 4b)\text{ cm}\) and \((5b + 2a)\text{ cm}\). Find its area and perimeter.
Answer:
Given:
Length (\(l\)) = \((3a - 4b)\text{ cm}\)
Breadth (\(b\)) = \((5b + 2a) = (2a + 5b)\text{ cm}\)

1. Area of rectangle:
\(\text{Area} = l \times b\)
\(\text{Area} = (3a - 4b)(2a + 5b)\)
\(= 3a(2a + 5b) - 4b(2a + 5b)\)
\(= 6a^2 + 15ab - 8ab - 20b^2\)
\(= 6a^2 + 7ab - 20b^2\text{ cm}^2\)

2. Perimeter of rectangle:
\(\text{Perimeter} = 2(l + b)\)
\(= 2[(3a - 4b) + (2a + 5b)]\)
\(= 2[(3a + 2a) + (-4b + 5b)]\)
\(= 2[5a + b]\)
\(= 10a + 2b\text{ cm}\)
In simple words: Multiply length and breadth to get the area, and add length and breadth then double it to get the perimeter.

Exam Tip: Include proper square units for area (\(\text{cm}^2\)) and linear units for perimeter (\(\text{cm}\)).