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If you are in search of NCERT solutions for Class 8 Maths Chapter 12 Factorisation, then you are at the right place. When it comes to Mathematics, it is a subject that holds value for students across various academic streams, be it science, biology, or commerce and having a very good understanding of basic math concepts will help you in achieving your goal and make it much more easier.
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NCERT Solutions for Class 8 Maths Chapter 12 Factorisation Ex 12.2
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Class 8 Maths Chapter 12 Factorisation Exercise 12.2
Exercise 12.2
1. Factorise the following expressions.
\[(i)\ a^2+8a+16\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (ii)\ p^2-10p+25\]
\[(iii)\ 25m^2+30m+9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (iv)\ 49y^2+84yz+36z^2\]
\[(v)\ 4x^2-8x+4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (vi)\ 121b^2-88bc+16c^2\]
\[(vii)\ (l+m)^2-4lm\ (Hint:\ Expand\ (l+m)^2\ first)\]
\[(viii)\ a^4+2a^2b^2+b^4\]
Solution:
\[(i)\ a^2+8a+16\ \]
\[U\sin g\ \ identity\ (x+y)^2=x^2+2xy+y^2\]
\[=a^2+2\times4\times a+4^2\ \ \]
\[=(a+4)^2\]
\[\]
\[(ii)\ p^2-10p+25\]
\[U\sin g\ \ identity\ (x-y)^2=x^2-2xy+y^2\]
\[=p^2-2\times5\times p+5^2\]
\[=(p-5)^2\]
\[\]
\[(iii)\ 25m^2+30m+9\]
\[U\sin g\ \ identity\ (x+y)^2=x^2+2xy+y^2\]
\[=(5m)^2+2\times5m\times3+3^2\]
\[=(5m+3)^2\]
\[\]
\[(iv)\ 49y^2+84yz+36z^2\]
\[U\sin g\ \ identity\ (x+y)^2=x^2+2xy+y^2\]
\[=(7y)^2+2\times7y\times6z+(6z)^2\]
\[=(7y+6z)^2\]
\[\]
\[(v)\ 4x^2-8x+4\]
\[U\sin g\ \ identity\ (x-y)^2=x^2-2xy+y^2\]
\[=(2x)^2-2\times4x+2^2\]
\[=(2x-2)^2\]
\[\]
\[(vi)\ 121b^2-88bc+16c^2\]
\[U\sin g\ \ identity\ (x-y)^2=x^2-2xy+y^2\]
\[=(11b)^2-2\times11b\times4c+(4c)^2\]
\[=(11b-4c)^2\]
\[\]
\[(vii)\ (l+m)^2-4lm\ \ (Hint:\ Expand\ (l+m)^2\ first)\]
\[Expanding\ (l+m)^2\ \ \ \left(u\sin g\ the\ identity\ (x+y)^2=x^2+2xy+y^2\right)\]
\[(l+m)^2-4lm=l^2+m^2+2lm-4lm\]
\[=l^2+m^2-2lm\ \ \left(U\sin g\ \ identity\ (x-y)^2=x^2-2xy+y^2\right)\]
\[=(l-m)^2\]
\[\]
\[(viii)\ a^4+2a^2b^2+b^4\]
\[U\sin g\ \ identity\ (x-y)^2=x^2-2xy+y^2\]
\[=(a^2)^2+2\times a^2\times b^2+(b^2)^2\ \ \ \ \ \ \ \ \]
\[=(a^2+b^2)^2\]
\[\]
2. Factorise.
\[(i)\ 4p^2-9q^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (ii)\ 63a^2-112b^2\]
\[(iii)\ 49x^2-36\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (iv)\ 16x^5-144x^3\]
\[(v)\ (l+m)^2-(l-m)^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (vi)\ 9x^2y^2-16\]
\[(vii)\ (x^2-2xy+y^2)-z^2\ \ \ \ \ \ \ \ \ \ \ (viii)\ 25a^2-4b^2+28bc-49c^2\]
Solution:
\[(i)\ 4p^2-9q^2\]
\[=(2p)^2-(3q)^2\ \ \ \left(U\sin g\ \ identity\ x^2-y^2=(x+y)(x-y)\right)\]
\[=(2p+3q)(2p-3q)\]
\[\]
\[(ii)\ 63a^2-112b^2\]
\[=7(9a^2-16b^2)\ \ \ \left(U\sin g\ \ identity\ x^2-y^2=(x+y)(x-y)\right)\]
\[=7((3a)^2-(4b)^2)\]
\[=7(3a+4b)(3a-4b)\]
\[\]
\[(iii)\ 49x^2-36\]
\[=(7x)^2-6^2\ \ \ \left(U\sin g\ \ identity\ x^2-y^2=(x+y)(x-y)\right)\]
\[=(7x+6)(7x-6)\]
\[\]
\[(iv)\ 16x^5-144x^3\]
\[=16x^3(x^2-9)\ \ \]
\[=16x^3(x^2-3^2)\ \ \ \left(U\sin g\ \ identity\ x^2-y^2=(x+y)(x-y)\right)\]
\[=16x^3(x+3)(x-3)\]
\[\]
\[(v)\ (l+m)^2-(l-m)^2\]
\[\left(U\sin g\ \ identity\ x^2-y^2=(x+y)(x-y)\right)\]
\[=\left[(l+m)-(l-m)\right]\left[(l+m)+(l-m)\right]\]
\[=(l+m-l+m)(l+m+l-m)\]
\[=(2m)(2l)\ =4ml\]
\[\]
\[(vi)\ 9x^2y^2-16\]
\[=(3xy)^2-4^2\ \ \ \left(U\sin g\ \ identity\ x^2-y^2=(x+y)(x-y)\right)\]
\[=(3xy+4)(3xy-4)\]
\[\]
\[(vii)\ (x^2-2xy+y^2)-z^2\ \ \ \ \]
\[=(x-y)^2-z^2\ \ \ \ \ \left(U\sin g\ \ identity\ (x-y)^2=x^2-2xy+y^2\right)\]
\[=\left(x-y+z\right)\left(x-y-z\right)\ \ \ \left(U\sin g\ \ identity\ x^2-y^2=(x+y)(x-y)\right)\]
\[\]
\[(viii)\ 25a^2-4b^2+28bc-49c^2\]
\[=25a^2-(4b^2-28bc+49c^2)\]
\[=(5a)^2-(2b)^2-2(2b)(7c)+(7c)^2\]
\[=(5a)^2-(2b-7c)^2\ \ \ \left(U\sin g\ identity\ (x-y)^2=x^2-2xy+y^2\right)\]
\[=(5a+2b-7c)(5a-2b+7c)\ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[\]
3. Factorise the expressions.
\[(i)\ ax^2+bx\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (ii)\ 7p^2+21q^2\]
\[(iii)\ 2x^3+2xy^2+2xz^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (iv)\ am^2+bm^2+bn^2+an^2\]
\[(v)\ (lm+l)+m+1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (vi)\ y(y+z)+9(y+z)\]
\[(vii)\ 5y^2-20y-8z+2yz\ \ \ \ \ \ \ \ \ \ \ \ \ \ (viii)10ab+4a+5b+2\]
\[(ix)\ 6xy-4y+6-9x\]
Solution:
\[(i)\ ax^2+bx=\ x(ax+b)\]
\[(ii)\ 7p^2+21q^2=7(p^2+3q^2)\]
\[(iii)\ 2x^2+2xy^2+2xz^2=2x(x^2+y^2+z^2)\]
\[(iv)\ am^2+bm^2+bn^2+an^2=m^2(a+b)+n^2(a+b)=(a+b)(m^2+n^2)\]
\[(v)\ (lm+l)+m+1=lm+m+l+1=m(l+1)+(l+1)=(m+1)(l+1)\]
\[(vi)\ y(y+z)+9(y+z)=(y+9)(y+z)\]
\[(vii)\ 5y^2-20y-8z+2yz\]
\[=5y(y-4)+2z(y-4)\]
\[=(y-4)(5y+2z)\]
\[(viii)\ 10ab+4a+5b+2=5b(2a+1)+2(2a+1)=(2a+1)(5b+2)\]
\[(ix)\ 6xy-4y+6-9x=6xy-9x-4y+6\]
\[=3x(2y-3)-2(2y-3)\]
\[=(2y-3)(3x-2)\]
\[\]
4. Factorise.
\[(i)\ a^4-b^4\]
\[(ii)\ p^4-81\]
\[(iii)\ x^4-(y+z)^4\]
\[(iv)\ x^4-(x-z)^4\]
\[(v)\ a^4-2a^2b^2+b^4\]
Solution:
\[(i)\ a^4-b^4\]
\[=(a^2)^2-(b^2)^2\ \ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[=(a^2-b^2)(a^2+b^2)\ \ \ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[=(a-b)(a+b)(a^2+b^2)\]
\[\]
\[(ii)\ p^4-81\]
\[=(p^2)^2-(9)^2\ \ \ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[=(p^2-9)(p^2+9)\]
\[=(p^2-3^2)(p^2+9)\ \ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[=(p-3)(p+3)(p^2+9)\]
\[\]
\[(iii)\ x^4-(y+z)^4\]
\[=\left(x^2\right)^2-\left[(y+z)^2\right]^2\ \ \ \ \ \ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[=\left[x^2-(y+z)^2\right]\left[x^2+(y+z)^2\right]\ \ \ \ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[=\left[(x-(y+z)\right]\left[(x+(y+z)\right]\left[x^2+(y+z)^2\right]\]
\[=(x-y-z)(x+y+z)\left[x^2+(y+z)^2\right]\]
\[\]
\[(iv)\ x^4-(x-z)^4\]
\[=\left(x^2\right)^2-\left[(x-z)^2\right]^2\ \ \ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[=\left[x^2-(x-z)^2\right]\left[x^2+(x-z)^2\right]\ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[=\left[x-(x-z)\right]\left[x+(x-z)\right]\left[x^2+(x-z)^2\right]\]
\[=z(2x-z)\left[(x^2+x^2-2xz+z^2)\right]\]
\[=z(2x-z)(2x^2-2xz+z^2)\]
\[\]
\[(v)\ a^4-2a^2b^2+b^4\]
\[=\left(a^2\right)^2-2a^2b^2+\left(b^2\right)^2\]
\[=(a^2-b^2)^{^2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(U\sin g\ identity\ (x-y)^2=x^2-2xy+y^2\right)\]
\[=\left[(a-b)(a+b)\right]^2\ \ \ \ \ \ \ \ \ \ \left(U\sin g\ Identity\ \ x^2-y^2=(x+y)(x-y)\right)\]
\[=(a-b)^2(a+b)^2\ \ \ \ \ \ \ \ \ \ \ \ \ \left(U\sin g\ identity\ \ \left(xy\right)^m=x^m.y^m\right)\]
\[\]
5. Factorise the following expressions.
\[(i)\ p^2+6p+8\]
\[(ii)\ q^2-10q+21\]
\[(iii)\ p^2+6p-16\]
Solution:
\[(i)\ p^2+6p+8\ \]
\[=p^2+\left(2+4\right)p+\left(2\times4\right)\]
\[=p^2+2p+4p+8\]
\[=p(p+2)+4(p+2)\]
\[=(p+2)(p+4)\]
\[\]
\[(ii)\ q^2-10q+21\]
\[=q^2-\left(7+3\right)q+\left(7\times3\right)\]
\[=q^2-3q-7q+21\]
\[=q(q-3)-7(q-3)\]
\[=(q-7)(q-3)\]
\[\]
\[(iii)\ p^2+6p-16\]
\[=p^2+\left(8-2\right)p-8\times2\]
\[=p^2+8p-2p-16\]
\[=p(p+8)-2(p+8)\]
\[=(p+8)(p-2)\]
\[\]
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