Question

Factorise the following:
$(i)6a + 6b$
$(ii)ax + bx$
$(iii)3{x^2} - 6{a^6}$
$(iv)9{x^2} + 3x$
$(v)12{x^2}y - 4x{y^2}$
$(vi)\dfrac{1}{2}x + \dfrac{1}{2}$
$(vii)cdm + cdt$
$(viii)36{a^2}{b^3} - 18{a^3}{b^2}$
$(ix)25{m^2}{n^3} - 5mn$
$(x)3ay + 3az$
$(xi)185a + 185b$
$(xii)28x - 14y$
$(xiii)ax - ay$
$(xiv)12{y^3} + 6{a^3}$
$(xv)3x + 9y$

Answer 1

Hint: Factorization means to take the common terms out of the given equation and then to write in the form of factors.

Complete step-by-step answer:
$(i)6a + 6b$
Taking 6 common, we get-
$6(a + b)$(Ans)
$(ii)ax + bx$
Taking x common, we get-
$x(a + b)$(Ans)
$(iii)3{x^2} - 6{a^6}$
Taking 3 common, we get-
$3({x^2} - 2{a^6}) \\
\Rightarrow 3(x - \sqrt 2 {a^3})(x + \sqrt 2 {a^3})\{ \because {c^2} - {d^2} = (c + d)(c - d)\} {\text{(Ans)}} \\$
$(iv)9{x^2} + 3x$
Taking 3x common, we get-
$3x(3x + 1)$(Ans)
$(v)12{x^2}y - 4x{y^2}$
Taking 4xy common, we get-
$4xy(3x - y)$(Ans)
$(vi)\dfrac{1}{2}x + \dfrac{1}{2}$
Taking 1/2 common, we get-
$\dfrac{1}{2}(x + 1)$(Ans)
$(vii)cdm + cdt$
Taking cd common, we get-
$cd(m + t)$(Ans)
$(viii)36{a^2}{b^3} - 18{a^3}{b^2}$
Taking $18{a^2}{b^2}$ common, we get-
$18{a^2}{b^2}(2b - a)$(Ans)
$(ix)25{m^2}{n^3} - 5mn$
Taking 5mn common, we get-
$5mn(5m{n^2} - 1)$(Ans)
$(x)3ay + 3az$
Taking 3a common, we get-
$3a(y + z)$(Ans)
$(xi)185a + 185b$
Taking 185 common, we get-
$185(a + b)$(Ans)
$(xii)28x - 14y$
Taking 14 common, we get-
$14(2x - y)$(Ans)
$(xiii)ax - ay$
Taking a common, we get-
$a(x - y)$(Ans)
$(xiv)12{y^3} + 6{a^3}$
Taking 6 common, we get-
$6(2{y^3} + {a^3})$
We can also write $6({(\sqrt[3]{2})^3}{y^3} + {a^3})$is of the form ${a^3} + {b^3} = (a + b)({a^2} + {b^2} - ab)$
So, the above expression will be-
$ = 6(\sqrt[3]{2}y + a)({y^2}{(2)^{\dfrac{2}{3}}} + {a^2} - ay{(2)^{\dfrac{1}{3}}})$(Ans)
$(xv)3x + 9y$
Taking 3 common, we get-
$3(x + 3y)$(Ans)

Note: Whenever such types of questions appear, where you have to perform factorisation, then always write down the given polynomial, and then search for the common term among all the terms of the polynomial and then simplify to the simplest form in terms of factors.