Ex.14.2 Q4 Factorization - NCERT Maths Class 8

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Question

 Factorise

(i) \(\begin{align}{a^4} - {b^4}\end{align}\)

(ii) \(\begin{align}{p^4} - 81\end{align}\)

(iii) \(\begin{align}{x^4} - {{(y + z)}^4}\end{align}\)

(vi) \(\begin{align}{x^4} - {{(x - z)}^4}\end{align}\)

(v) \(\begin{align} {a^4} - 2{a^2}{b^2} + {b^4} \end{align}\)

Text Solution

What is known:

Algebraic expression.

What is unknown:

Factorisation of the algebraic expression.

Reasoning: Use identity:

\[\begin{align}  & {{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}} \\  & {{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right) \\ \end{align}\]

Steps:

\(\begin{align}{\rm{(i)}}\quad & {a^4} - {b^4} \\ &= {{\left( {{a^2}} \right)}^2} - {{\left( {{b^2}} \right)}^2}\\&= \left( {{a^2} - {b^2}} \right)\left( {{a^2} + {b^2}} \right)\\&= (a - b)(a + b)\left( {{a^2} + {b^2}} \right)\end{align}\)

\(\begin{align}{\rm{(ii)}}\quad & {p^4} - 81 \\ &= {{\left( {{p^2}} \right)}^2} - {{(9)}^2}\\&= \left( {{p^2} - 9} \right)\left( {{p^2} + 9} \right)\\&= \left[ {{{(p)}^2} - {{(3)}^2}} \right]\left( {{p^2} + 9} \right)\\&= (p - 3)(p + 3)\left( {{p^2} + 9} \right)\end{align}\)

\(\begin{align}\left( {{\rm{iii}}} \right) \quad & {x^4} - {{(y + z)}^4} \\ &= {{\left( {{x^2}} \right)}^2} - {{\left[ {{{(y + z)}^2}} \right]}^2}\\&= \begin{Bmatrix} \left[ {{x^2} - {{(y + z)}^2}} \right] \\ \left[ {{x^2} + {{(y + z)}^2}} \right] \end{Bmatrix} \\ &= \begin{Bmatrix}  [x - (y + z)] \\ [x + (y + z)] \\ \left[ {{x^2} + {{(y + z)}^2}} \right] \end{Bmatrix} \\&= \begin{Bmatrix} (x - y - z) \\ (x + y + z)\\ \left[ {{x^2} + {{(y + z)}^2}} \right] \end{Bmatrix} \end{align}\)

\(\begin{align} \left( \rm{iv} \right)\quad & {x^{\rm{4}}} - {{\left( {x - z} \right)}^{\rm{4}}} \\ &= {{\left( {{x^2}} \right)}^2} - {{\left[ {{{\left( {x - z} \right)}^2}} \right]}^2}\\ &=\begin{Bmatrix}  \left[ {{x^2} - {{\left( {x - z} \right)}^{\rm{2}}}} \right] \\ \left[ {{x^{\rm{2}}} + {{\left( {x - z} \right)}^{\rm{2}}}} \right] \end{Bmatrix} \\ &= \begin{Bmatrix} \left[ {x - \left( {x - z} \right)} \right] \\ \left[ {x + \left( {x - z} \right)} \right] \\ \left[ {{x^{\rm{2}}} + {{\left( {x - z} \right)}^{\rm{2}}}} \right] \end{Bmatrix} \\ &= \begin{Bmatrix} z\left( {2x - z} \right) \\ \begin{bmatrix}  x^{\rm{2}} + x^{\rm{2}} \\ - 2xz + {z^{\rm{2}}} \end{bmatrix} \end{Bmatrix} \\&= \begin{Bmatrix} z \left( {{\rm{2}}x - z} \right) \\ \left( {{\rm{2}}{x^{\rm{2}}} - {\rm{2}}xz + {z^{\rm{2}}}} \right)\end{Bmatrix} \end{align}\)

\(\begin{align}({\rm{v}})\quad & {a^4} - 2{a^2}{b^2} + {b^4} \\ &= \begin{Bmatrix} \left( {{a^2}} \right)^2 - 2\left( {{a^2}} \right)\left( {{b^2}} \right) \\ + \left( {{b^2}} \right)^2 \end{Bmatrix} \\&= {{\left( {{a^2} - {b^2}} \right)}^2}\\&= {{\left[ {(a - b)(a + b)} \right]}^2}\\&= {{(a - b)}^2}{{(a + b)}^2}\end{align}\)

  
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