Question

8(a+b)³+c³ resolve in to factor
step by step answer
answer must be :
(2a+2b+c)(4a²+4b²+8ab-2ac-2bc+c²)​

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: 8 {(a + b)}^{3} + {c}^{3}$

can be rewritten as

$\rm \: = 2 \times 2 \times 2 \times {(a + b)}^{3} + {c}^{3}$

$\rm \: = {2}^{3} \times {(a + b)}^{3} + {c}^{3}$

$\rm \: ={(2[a + b])}^{3} + {c}^{3}$

$\rm \: ={(2a + 2b)}^{3} + {c}^{3}$

We know,

$\boxed{ \rm{ \: {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2}) \: }}$

So, here

$\rm \: x = 2a + 2b$

$\rm \: y = c$

So, on substituting the values in above identity, we get

$\rm \: = (2a + 2b + c)[ {(2a + 2b)}^{2} - (2a + 2b)c + {c}^{2}]$

$\rm \: = (2a + 2b + c)[ {(2a)}^{2} + {(2b)}^{2} + 2(2a)(2b) - 2ac - 2bc + {c}^{2}]$

$\rm \: = (2a + 2b + c)[ {4a}^{2} + {4b}^{2} + 8ab - 2ac - 2bc + {c}^{2}]$

Hence,

$\rm\implies \: 8 {(a + b)}^{3} \: + \: {c}^{3} \\ \\ \rm \: = (2a + 2b + c)[ {4a}^{2} + {4b}^{2} + 8ab - 2ac - 2bc + {c}^{2}]$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$