| Factorise the expressions and divide them as directed. |
| (a) \[\left( {{y}^{2}}+7y+10 \right)\div \left( y+5 \right)\] |
| (b) \[\left( {{m}^{2}}-14m-32 \right)\div \left( \text{m}+2 \right)\] |
| (c) \[\left( 5{{p}^{2\text{ }}}-25p+20 \right)\div \left( p-1 \right)\] |
| (d) \[4yz\left( {{z}^{2}}+\text{ }6z-16 \right)\div \text{ }2y\text{ }\left( z+8 \right)\] |
Answer:
(a) \[\left( {{y}^{2}}+7y+10 \right)\div \left( y+5 \right)\] Dividend \[=\text{ }{{y}^{2}}+7y+10\] \[={{y}^{2}}+\left( 5+2 \right)y+10\] \[={{y}^{2}}+5y+2y+10\] \[=y\left( y+5 \right)+2\left( y+5 \right)\] \[=\left( y+5 \right)\left( y+2 \right)\] Thus, \[({{y}^{2}}+7y+10)\div (y+5)=\frac{(y+5)(y+2)}{(y+5)}=y+2\] (b) \[({{m}^{2}}14m32)\div (m+2)\] Dividend = m2 - 14m - 32 \[={{m}^{2}}\left( 162 \right)\text{ }m32\] \[=\text{ }{{m}^{2}}16m+2m32\] \[=\text{ }m\left( m16 \right)+2\left( m16 \right)\] \[=\left( m16 \right)\text{ }\left( m+2 \right)\] Thus, \[\text{(}{{m}^{2}}14m32)\div \left( m+2 \right)\] \[=\frac{(m-16)(m+2)}{(m+2)}\] \[=\text{ }m\text{ }\text{ }16\] (c) \[\left( 5{{p}^{2}}25p+20 \right)\div \left( p1 \right)\] Dividend \[=\text{ }5{{p}^{2}}25p+20\] \[=5{{p}^{2}}\left( 20+5 \right)p+20\] \[=\text{ }5{{p}^{2}}20p5p+20\] \[=5p\left( p4 \right)5\left( p4 \right)\] \[=\left( p4 \right)\left( 5p5 \right)\] \[=5\left( p4 \right)\left( p1 \right)\] Thus, \[\left( 5{{p}^{2}}25p+20 \right)\div \left( p1 \right)\] \[=\frac{5(p-4)(p-1)}{(p-1)}\] \[=5(p-4)\] (d) \[4yz({{z}^{2}}+6z16)\div 2y(z+8)\] Dividend \[=\text{ }4yz({{z}^{2}}+6z16)\] \[=4yz[{{z}^{2\text{ }}}+(82)\text{ }z16]\] \[=\text{ }4yz\text{ }\!\![\!\!\text{ }{{z}^{2}}+8z2z16]\] \[=\text{ }4yz=[z(z+8)-2(z+8)]\] \[=\text{ }4yz\text{ (}z+8)\text{ (}z2)\] Thus, \[4yz({{z}^{2}}+6z16)\div 2y(z+8)\] \[=\frac{4yz(z+8)(z-2)}{2y(z+8)}\] \[=2z(z-2)\]
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