Q1. Factorize:
(i) $15x + 5$
(ii) $a^3 - a^2 + a$
(iii) $3x^2 + 6x^3$
(iv) $3a^2 - 9ab$
(v) $2a^5 - 6a^3$
Show Answer
(i) $15x + 5$
Common Factor = 5
$\Rightarrow 5(\dfrac{15x}{5} + \dfrac{5}{5})$
$\Rightarrow 5(3x + 1)$
$\therefore 15x + 5$ Ans
(ii) $a^3 - a^2 + a$
Common Factor = a
$\Rightarrow a(\dfrac{a^3}{a} - \dfrac{a^2}{a} + \dfrac{a}{a})$
$\therefore a(a^2 - a + 1)$ Ans
(iii) $3x^2 + 6x^3$
Common Factor = $3x^2$
$\Rightarrow 3x^2(\dfrac{3x^2}{3x^2} + \dfrac{6x^3}{3x^2})$
$\therefore 3x^2(1 + 2x)$ Ans
(iv) $3a^2 - 9ab$
Common Factor = $3a$
$\therefore 3a(a - 3b)$ Ans
(v) $2a^5 - 6a^3$
Common Factor = $2a^3$
$\therefore 2a^3(2a^2 - 3)$ Ans
Common Factor = 5
$\Rightarrow 5(\dfrac{15x}{5} + \dfrac{5}{5})$
$\Rightarrow 5(3x + 1)$
$\therefore 15x + 5$ Ans
(ii) $a^3 - a^2 + a$
Common Factor = a
$\Rightarrow a(\dfrac{a^3}{a} - \dfrac{a^2}{a} + \dfrac{a}{a})$
$\therefore a(a^2 - a + 1)$ Ans
(iii) $3x^2 + 6x^3$
Common Factor = $3x^2$
$\Rightarrow 3x^2(\dfrac{3x^2}{3x^2} + \dfrac{6x^3}{3x^2})$
$\therefore 3x^2(1 + 2x)$ Ans
(iv) $3a^2 - 9ab$
Common Factor = $3a$
$\therefore 3a(a - 3b)$ Ans
(v) $2a^5 - 6a^3$
Common Factor = $2a^3$
$\therefore 2a^3(2a^2 - 3)$ Ans
Q2. Factorize:
(i) $4a^2 - 8ab$
(ii) $2x^3b^2 - 4x^5b^4$
(iii) $15x^4y^3 - 20x^3y$
(iv) $a^3b - a^2b^2 - b^3$
(v) $6x^2y + 9xy^2 + 4y^3$
Show Answer
(i) $4a^2 - 8ab$
Common Factor = $4a$
$\Rightarrow 4a(\dfrac{4a^2}{4a} - \dfrac{8ab}{4a})$
$\therefore 4a(a - 2b)$ Ans
(ii) $2x^3b^2 - 4x^5b^4$
Common Factor = $2x^3b^2$
$\Rightarrow 2x^3b^2(\dfrac{2x^3b^2}{2x^3b^2} - \dfrac{4x^5b^4}{2x^3b^2})$
$\therefore 2x^3b^2(1 - 2x^2b^2)$ Ans
(iii) $15x^4y^3 - 20x^3y$
Common Factor = $5x^3y$
$\Rightarrow 5x^3y(\dfrac{15x^4y^3}{5x^3y} - \dfrac{20x^3y}{5x^3y})$
$\therefore 5x^3y(3xy^2 - 4)$ Ans
(iv) $a^3b - a^2b^2 - b^3$
Common Factor = b
$\Rightarrow b(\dfrac{a^3b}{b} - \dfrac{a^2b^2}{b} - \dfrac{b^3}{b})$
$\therefore b(a^3 - a^2b - b^2)$ Ans
(v) $6x^2y + 9xy^2 + 4y^3$
Common Factor = y
$\Rightarrow y(\dfrac{6x^2y}{y} + \dfrac{9xy^2}{y} + \dfrac{4y^3}{y})$
$\therefore y(6x^2 + 9xy + 4y^2)$ Ans
Common Factor = $4a$
$\Rightarrow 4a(\dfrac{4a^2}{4a} - \dfrac{8ab}{4a})$
$\therefore 4a(a - 2b)$ Ans
(ii) $2x^3b^2 - 4x^5b^4$
Common Factor = $2x^3b^2$
$\Rightarrow 2x^3b^2(\dfrac{2x^3b^2}{2x^3b^2} - \dfrac{4x^5b^4}{2x^3b^2})$
$\therefore 2x^3b^2(1 - 2x^2b^2)$ Ans
(iii) $15x^4y^3 - 20x^3y$
Common Factor = $5x^3y$
$\Rightarrow 5x^3y(\dfrac{15x^4y^3}{5x^3y} - \dfrac{20x^3y}{5x^3y})$
$\therefore 5x^3y(3xy^2 - 4)$ Ans
(iv) $a^3b - a^2b^2 - b^3$
Common Factor = b
$\Rightarrow b(\dfrac{a^3b}{b} - \dfrac{a^2b^2}{b} - \dfrac{b^3}{b})$
$\therefore b(a^3 - a^2b - b^2)$ Ans
(v) $6x^2y + 9xy^2 + 4y^3$
Common Factor = y
$\Rightarrow y(\dfrac{6x^2y}{y} + \dfrac{9xy^2}{y} + \dfrac{4y^3}{y})$
$\therefore y(6x^2 + 9xy + 4y^2)$ Ans
Q3. Factorize:
(i) $17a^6b^8 - 34a^4b^5 + 51a^2b^4$
(ii) $3x^5y - 27x^4y^2 + 12x^3y^3$
(iii) $x^2(a - b) - y^2(a - b) + z^2(a - b)$
(iv) $2b(2a + b) - 3c(2a + b)$
(v) $(x + y)(a + b) + (x - y)(a + b)$
(vi) $(a - b)^2 - 2(a - b)$
Show Answer
(i) $17a^6b^8 - 34a^4b^5 + 51a^2b^4$
Common Factor = $17a^2b^4$
$\Rightarrow 17a^2b^4(\dfrac{17a^6b^8}{17a^2b^4} - \dfrac{34a^4b^5}{17a^2b^4} + \dfrac{51a^2b^4}{17a^2b^4})$
$\therefore 17a^2b^4(a^4b^4 - 2a^2b + 3)$ Ans
(ii) $3x^5y - 27x^4y^2 + 12x^3y^3$
Common Factor = $3x^3y$
$\Rightarrow 3x^3y(\dfrac{3x^5y}{3x^3y} - \dfrac{27x^4y^2}{3x^3y} + \dfrac{12x^3y^3}{3x^3y})$
$\therefore 3x^3y(x^2 - 9xy + 4y^2)$ Ans
(iii) $x^2(a - b) - y^2(a - b) + z^2(a - b)$
Common Factor = $a - b$
$\Rightarrow a - b(\dfrac{x^2(a - b)}{a - b} - \dfrac{y^2(a - b)}{a - b} + \dfrac{z^2(a - b)}{a - b})$
$\therefore (a - b)(x^2 - y^2 + z^2)$ Ans
(iv) $2b(2a + b) - 3c(2a + b)$
Common Factor = $2a + b$
$\Rightarrow 2a + b(\dfrac{2b(2a + b)}{2a + b} - \dfrac{3c(2a + b)}{2a + b})$
$\therefore (2a + b)(2b - 3c)$ Ans
(v) $(x + y)(a + b) + (x - y)(a + b)$
Common Factor = $a + b$
$\Rightarrow a + b(\dfrac{(x + y)(a + b)}{a + b} + \dfrac{(x - y)(a + b)}{a + b})$
$\Rightarrow (a + b)(x + y) + (x - y)$
$\Rightarrow (a + b)(x + y + x - y)$
$\therefore 2x(a + b)$ Ans
(vi) $(a - b)^2 - 2(a - b)$
Taking $a - b = x$, we have: $x^2 - 2x$
$\Rightarrow x^2 - 2x = x(x - 2)$
$\therefore (a - b)(a - b - 2)$ Ans
Common Factor = $17a^2b^4$
$\Rightarrow 17a^2b^4(\dfrac{17a^6b^8}{17a^2b^4} - \dfrac{34a^4b^5}{17a^2b^4} + \dfrac{51a^2b^4}{17a^2b^4})$
$\therefore 17a^2b^4(a^4b^4 - 2a^2b + 3)$ Ans
(ii) $3x^5y - 27x^4y^2 + 12x^3y^3$
Common Factor = $3x^3y$
$\Rightarrow 3x^3y(\dfrac{3x^5y}{3x^3y} - \dfrac{27x^4y^2}{3x^3y} + \dfrac{12x^3y^3}{3x^3y})$
$\therefore 3x^3y(x^2 - 9xy + 4y^2)$ Ans
(iii) $x^2(a - b) - y^2(a - b) + z^2(a - b)$
Common Factor = $a - b$
$\Rightarrow a - b(\dfrac{x^2(a - b)}{a - b} - \dfrac{y^2(a - b)}{a - b} + \dfrac{z^2(a - b)}{a - b})$
$\therefore (a - b)(x^2 - y^2 + z^2)$ Ans
(iv) $2b(2a + b) - 3c(2a + b)$
Common Factor = $2a + b$
$\Rightarrow 2a + b(\dfrac{2b(2a + b)}{2a + b} - \dfrac{3c(2a + b)}{2a + b})$
$\therefore (2a + b)(2b - 3c)$ Ans
(v) $(x + y)(a + b) + (x - y)(a + b)$
Common Factor = $a + b$
$\Rightarrow a + b(\dfrac{(x + y)(a + b)}{a + b} + \dfrac{(x - y)(a + b)}{a + b})$
$\Rightarrow (a + b)(x + y) + (x - y)$
$\Rightarrow (a + b)(x + y + x - y)$
$\therefore 2x(a + b)$ Ans
(vi) $(a - b)^2 - 2(a - b)$
Taking $a - b = x$, we have: $x^2 - 2x$
$\Rightarrow x^2 - 2x = x(x - 2)$
$\therefore (a - b)(a - b - 2)$ Ans
Q4. Factorize:
(i) $2a(p^2 + q^2) + 4b(p^2 + q^2)$
(ii) $12abc - 6a^2b^2c^2 + 3a^3b^3c^3$
(iii) $4x(3x - 2y) - 2y(3x - 2y)$
(iv) $(a + 2b)(3a + b) - (a + b)(a + 2b) + (a + 2b)^2$
(v) $6xy(a^2 + b^2) + 8yz(a^2 + b^2) - 10xz(a^2 + b^2)$
Show Answer
(i) $2a(p^2 + q^2) + 4b(p^2 + q^2)$
Taking $p^2 + q^2 = x$, we have: $2ax + 4bx$
$\Rightarrow 2ax + 4bx = 2x(a + 2b)$
$\therefore 2(p^2 + q^2)(a + 2b)$ Ans
(ii) $12abc - 6a^2b^2c^2 + 3a^3b^3c^3$
Common Factor = $3abc$
$\Rightarrow 3abc(\dfrac{(12abc)}{3abc} - \dfrac{6a^2b^2c^2}{3abc} + \dfrac{3a^3b^3c^3}{3abc})$
$\therefore 3abc(4 - 2abc + a^2b^2c^2)$ Ans
(iii) $4x(3x - 2y) - 2y(3x - 2y)$
Common Factor = $2(3x - 2y)$
$\Rightarrow 2(3x - 2y)(\dfrac{4x(3x - 2y)}{2(3x - 2y)} - \dfrac{2y(3x - 2y)}{2(3x - 2y)})$
$\therefore 2(3x - 2y)(2x - y)$ Ans
(iv) $(a + 2b)(3a + b) - (a + b)(a + 2b) + (a + 2b)^2$
Common Factor = $a + 2b$
$\Rightarrow a + 2b(\dfrac{(a + 2b)(3a + b)}{a + 2b} - \dfrac{(a + b)(a + 2b)}{a + 2b} + \dfrac{a^2 + 4b^2}{a + 2b})$
$\Rightarrow (a + 2b)((3a + b) - (a + b) + (a + 2b))$
$\Rightarrow (a + 2b)(3a + b - a - b + a + 2b)$
$\therefore (a + 2b)(3a + 2b)$ Ans
(v) $6xy(a^2 + b^2) + 8yz(a^2 + b^2) - 10xz(a^2 + b^2)$
Common Factor = $2(a^2 + b^2)$
$\Rightarrow 2(a^2 + b^2)(\dfrac{6xy(a^2 + b^2)}{2(a^2 + b^2)} + \dfrac{8yz(a^2 + b^2)}{2(a^2 + b^2)} - \dfrac{10xz(a^2 + b^2)}{2(a^2 + b^2)})$
$\therefore 2(a^2 + b^2)(3xy + 4yz - 5xz)$ Ans
Taking $p^2 + q^2 = x$, we have: $2ax + 4bx$
$\Rightarrow 2ax + 4bx = 2x(a + 2b)$
$\therefore 2(p^2 + q^2)(a + 2b)$ Ans
(ii) $12abc - 6a^2b^2c^2 + 3a^3b^3c^3$
Common Factor = $3abc$
$\Rightarrow 3abc(\dfrac{(12abc)}{3abc} - \dfrac{6a^2b^2c^2}{3abc} + \dfrac{3a^3b^3c^3}{3abc})$
$\therefore 3abc(4 - 2abc + a^2b^2c^2)$ Ans
(iii) $4x(3x - 2y) - 2y(3x - 2y)$
Common Factor = $2(3x - 2y)$
$\Rightarrow 2(3x - 2y)(\dfrac{4x(3x - 2y)}{2(3x - 2y)} - \dfrac{2y(3x - 2y)}{2(3x - 2y)})$
$\therefore 2(3x - 2y)(2x - y)$ Ans
(iv) $(a + 2b)(3a + b) - (a + b)(a + 2b) + (a + 2b)^2$
Common Factor = $a + 2b$
$\Rightarrow a + 2b(\dfrac{(a + 2b)(3a + b)}{a + 2b} - \dfrac{(a + b)(a + 2b)}{a + 2b} + \dfrac{a^2 + 4b^2}{a + 2b})$
$\Rightarrow (a + 2b)((3a + b) - (a + b) + (a + 2b))$
$\Rightarrow (a + 2b)(3a + b - a - b + a + 2b)$
$\therefore (a + 2b)(3a + 2b)$ Ans
(v) $6xy(a^2 + b^2) + 8yz(a^2 + b^2) - 10xz(a^2 + b^2)$
Common Factor = $2(a^2 + b^2)$
$\Rightarrow 2(a^2 + b^2)(\dfrac{6xy(a^2 + b^2)}{2(a^2 + b^2)} + \dfrac{8yz(a^2 + b^2)}{2(a^2 + b^2)} - \dfrac{10xz(a^2 + b^2)}{2(a^2 + b^2)})$
$\therefore 2(a^2 + b^2)(3xy + 4yz - 5xz)$ Ans