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Math Notes For Class 8 Chapter 3 Ex 3.2 Solved & Quiz

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Math Notes For Class 8 Chapter 3 Ex 3.2 Solved & Quiz

Official Academic Study Notes • Published: October 10, 2026 • Free Printable Resource

What are the Core Formulas and Definitions for this Exercise?

In Class 8 Mathematics (New 2025-2026 SNC Syllabus), Chapter 3 focuses on algebraic expressions and polynomials. Exercise 3.2 specifically covers basic operations on polynomials: Addition, Subtraction, Multiplication, and Division.

Polynomial: An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Like Terms: Terms that have identical variables raised to the exact same powers. Only like terms can be added or subtracted directly by operating on their numerical coefficients.
Polynomial Addition & Subtraction: Performed by grouping like terms together and computing their net coefficients. When subtracting a polynomial, the signs of all terms in the subtracted polynomial must be inverted.
Polynomial Multiplication: Uses the distributive law $a(b+c) = ab + ac$ along with the product law of exponents ($x^m \cdot x^n = x^{m+n}$). Every term of the first polynomial must be multiplied by every term of the second.
Polynomial Division: Executed using the algebraic long division method where dividend polynomial $P(x)$ is divided by divisor $D(x)$ to yield quotient $Q(x)$ and remainder $R(x)$.

Operation / Rule	Mathematical Formula / Expression	Key Requirement / Property
Addition of Like Terms	$a x^n + b x^n = (a + b) x^n$	Variables and exponents must match exactly.
Subtraction Inversion	$P(x) - Q(x) = P(x) + [-Q(x)]$	Distribute $-1$ into every term of $Q(x)$.
Product Law of Exponents	$x^m \cdot x^n = x^{m+n}$	Bases must be identical; exponents are added.
Quotient Law of Exponents	$\frac{x^m}{x^n} = x^{m-n} \quad (x \neq 0, m \ge n)$	Bases must be identical; exponents are subtracted.
Division Identity	$P(x) = D(x) \cdot Q(x) + R(x)$	$\text{Degree}(R(x)) < \text{Degree}(D(x))$.

How to Solve All Exercise Questions Step-by-Step?

Question 1 (Part i): Add the polynomials: $P(x) = 3x^3 + 2x^2 - 5x + 7$ and $Q(x) = 2x^3 - 4x^2 + 8x - 3$.

Solution:
Write the two polynomials inside brackets and combine like terms:
$$P(x) + Q(x) = (3x^3 + 2x^2 - 5x + 7) + (2x^3 - 4x^2 + 8x - 3)$$ Group like terms together:
$$= (3x^3 + 2x^3) + (2x^2 - 4x^2) + (-5x + 8x) + (7 - 3)$$ Perform addition/subtraction of coefficients:
$$= (3+2)x^3 + (2-4)x^2 + (-5+8)x + (7-3)$$ $$= 5x^3 - 2x^2 + 3x + 4$$ Answer: $$5x^3 - 2x^2 + 3x + 4$$

Question 1 (Part ii): Add the polynomials: $A(x,y) = 4x^2 - 3xy + 2y^2$ and $B(x,y) = -2x^2 + 5xy - 6y^2$.

Solution:
Write the sum of both polynomials:
$$A(x,y) + B(x,y) = (4x^2 - 3xy + 2y^2) + (-2x^2 + 5xy - 6y^2)$$ Group like terms:
$$= (4x^2 - 2x^2) + (-3xy + 5xy) + (2y^2 - 6y^2)$$ Combine coefficients:
$$= (4 - 2)x^2 + (-3 + 5)xy + (2 - 6)y^2$$ $$= 2x^2 + 2xy - 4y^2$$ Answer: $$2x^2 + 2xy - 4y^2$$

Question 1 (Part iii): Add the polynomials: $L = \frac{1}{2}x^2 + 3x - 4$ and $M = \frac{3}{2}x^2 - 5x + 9$.

Solution:
Combine like terms directly:
$$L + M = \left(\frac{1}{2}x^2 + \frac{3}{2}x^2\right) + (3x - 5x) + (-4 + 9)$$ $$= \left(\frac{1 + 3}{2}\right)x^2 + (3 - 5)x + (5)$$ $$= \left(\frac{4}{2}\right)x^2 - 2x + 5 = 2x^2 - 2x + 5$$ Answer: $$2x^2 - 2x + 5$$

Question 2 (Part i): Subtract $Q(x) = 2x^2 - 3x + 5$ from $P(x) = 5x^2 + 4x - 2$.

Solution:
Express as $P(x) - Q(x)$:
$$P(x) - Q(x) = (5x^2 + 4x - 2) - (2x^2 - 3x + 5)$$ Distribute the negative sign through the second polynomial:
$$= 5x^2 + 4x - 2 - 2x^2 + 3x - 5$$ Group like terms:
$$= (5x^2 - 2x^2) + (4x + 3x) + (-2 - 5)$$ $$= 3x^2 + 7x - 7$$ Answer: $$3x^2 + 7x - 7$$

Question 2 (Part ii): Subtract $B(a,b) = 3a^2 - 7ab + 4b^2$ from $A(a,b) = 8a^2 + 2ab - 3b^2$.

Solution:
Setup subtraction: $A(a,b) - B(a,b)$:
$$= (8a^2 + 2ab - 3b^2) - (3a^2 - 7ab + 4b^2)$$ Invert all signs inside the second bracket:
$$= 8a^2 + 2ab - 3b^2 - 3a^2 + 7ab - 4b^2$$ Group like terms:
$$= (8a^2 - 3a^2) + (2ab + 7ab) + (-3b^2 - 4b^2)$$ $$= 5a^2 + 9ab - 7b^2$$ Answer: $$5a^2 + 9ab - 7b^2$$

Question 2 (Part iii): Subtract $N(x) = x^3 - 4x^2 + 6x - 1$ from $M(x) = 4x^3 + 2x^2 - 3x + 8$.

Solution:
Setup subtraction: $M(x) - N(x)$:
$$= (4x^3 + 2x^2 - 3x + 8) - (x^3 - 4x^2 + 6x - 1)$$ Distribute negative sign:
$$= 4x^3 + 2x^2 - 3x + 8 - x^3 + 4x^2 - 6x + 1$$ Group like terms:
$$= (4x^3 - x^3) + (2x^2 + 4x^2) + (-3x - 6x) + (8 + 1)$$ $$= 3x^3 + 6x^2 - 9x + 9$$ Answer: $$3x^3 + 6x^2 - 9x + 9$$

Question 3 (Part i): Multiply $(2x + 3)$ by $(3x^2 - 4x + 5)$.

Solution:
Apply the distributive property:
$$(2x + 3)(3x^2 - 4x + 5) = 2x(3x^2 - 4x + 5) + 3(3x^2 - 4x + 5)$$ Multiply term-by-term using product law of exponents ($x \cdot x^n = x^{n+1}$):
$$= (2x \cdot 3x^2