Ex.9.4 Q3 Algebraic Expressions and Identities - NCERT Maths Class 8

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Question

Simplify.

i) \(\left( {{x^2} - 5} \right){\rm{ }}\left( {x{\rm{ }} + 5} \right) + 25\)

ii) \(\left( {{a^2} + 5} \right)\left( {{b^3} + 3} \right){\rm{ }} + 5\)

iii) \(\left( {t + {s^2}} \right)\left( {{t^2} - s} \right)\)

iv) \(\begin{bmatrix} \left( {a{\rm{ }} + {\rm{ }}b} \right)\left( {c{\rm{ }} - {\rm{ }}d} \right) \\+ {\rm{ }}\left( {a{\rm{ }} - {\rm{ }}b} \right)\left( {c{\rm{ }} + {\rm{ }}d} \right)\\+ {\rm{ }}2{\rm{ }}\left( {ac{\rm{ }} + {\rm{ }}bd} \right)\end{bmatrix}\)

v) \(\begin{bmatrix}\left( {x{\rm{ }} + {\rm{ }}y} \right)\left( {2x{\rm{ }} + {\rm{ }}y} \right)\\+ {\rm{ }}\left( {x{\rm{ }} + {\rm{ }}2y} \right)\left( {x{\rm{ }} - {\rm{ }}y} \right)\end{bmatrix}\)

vi) \(\left( {x{\rm{ }} + {\rm{ }}y} \right)\left( {{x^2}{\rm{ }} - {\rm{ }}xy{\rm{ }} + {\rm{ }}{y^2}} \right)\)

vii) \(\begin{bmatrix}\left( {1.5x{\rm{ }} - {\rm{ }}4y} \right)\left( {1.5x{\rm{ }} + {\rm{ }}4y{\rm{ }} + {\rm{ }}3} \right){\rm{ }}\\ - {\rm{ }}4.5x{\rm{ }} + {\rm{ }}12y \end{bmatrix}\)

viii) \(\left( {a{\rm{ }} + b + {\rm{ }}c} \right)\left( {a{\rm{ }} + b{\rm{ }} - {\rm{ }}c} \right)\)

Text Solution

What is known?

Expressions

What is unknown?

Simplification

Steps:

i) \(\left( {{x^2} - 5} \right){\rm{ }}\left( {x{\rm{ }} + 5} \right) + 25\)

\[\begin{align}&= \begin{bmatrix} {x^2}\left( {x + 5} \right) - \\ 5\left( {x + 5} \right) + 25 \end{bmatrix} \\ &= \begin{bmatrix}{x^3} + 5{x^2} - \\  5x - 25 + 25 \end{bmatrix} \\&={x^3} + 5{x^2} - 5x\end{align}\]

ii) \(\left( {{a^2} + 5} \right)\left( {{b^3} + 3} \right){\rm{ }} + 5\)

\[\begin{align}&=\begin{bmatrix} {a^2}\left( {{b^3} + 3} \right)+ \\  5\left( {{b^3} + 3} \right) + 5 \end{bmatrix} \\&= \begin{bmatrix} {a^2}{b^3} + 3{a^2} +  \\ 5{b^3} + 15 + 5 \end{bmatrix} \\&= \begin{bmatrix} {a^2}{b^3}+ 3{a^2} + \\  5{b^3} + 20 \end{bmatrix} \end{align}\]

iii) \(\left( {t + {s^2}} \right)\left( {{t^2} - s} \right)\)

\[\begin{align}&= {\rm{ }}t{\rm{ }}\left( {{t^2} - s} \right) + {s^2}\left( {{t^2} - s} \right)\\&= {\rm{ }}{t^3} - st + {\rm{ }}{s^2}{t^2} - {s^3}\end{align}\]

iv)

 \(\begin{bmatrix} \left( {a{\rm{ }} + {\rm{ }}b} \right)\left( {c{\rm{ }} - {\rm{ }}d} \right) \\+ {\rm{ }}\left( {a{\rm{ }} - {\rm{ }}b} \right)\left( {c{\rm{ }} + {\rm{ }}d} \right)\\+ {\rm{ }}2{\rm{ }}\left( {ac{\rm{ }} + {\rm{ }}bd} \right)\end{bmatrix}\)

\[\begin{align}&=\begin{bmatrix}a\left( {c - d} \right)+ b\left( {c - d} \right) \\ + a\left( {c + d} \right) - b\left( {c + d} \right)\\  + 2\left( {ac + bd} \right)\end{bmatrix} \\ &= \begin{bmatrix}ac - ad + bc - bd \\  + ac + ad - bc \\ - bd + 2ac + 2bd \end{bmatrix}\\ &=\begin{bmatrix}\left( {ac + ac+2ac} \right) \\  + \left( {ad - ad} \right) +\left( {bc - bc} \right) \\  + \left( {2bd - bd - bd} \right) \end{bmatrix}\\&= 4ac\end{align}\]

v)

\(\begin{bmatrix}\left( {x{\rm{ }} + {\rm{ }}y} \right)\left( {2x{\rm{ }} + {\rm{ }}y} \right)\\+ {\rm{ }}\left( {x{\rm{ }} + {\rm{ }}2y} \right)\left( {x{\rm{ }} - {\rm{ }}y} \right)\end{bmatrix}\)

\[\begin{align}&= \begin{bmatrix} x \left( {2x + y} \right)+y \left( {2x + y} \right) \\ + x \left( {x - y} \right) + 2y \left( {x - y} \right)\end{bmatrix} \\&=\begin{bmatrix} 2{x^2} +  xy  +  2xy \\ +{y^2} + {x^2} -xy \\ + 2xy - 2{y^2}\end{bmatrix}\\ &= \begin{bmatrix} \left( {2{x^2} +{x^2}} \right)+ \left( {{y^2} -2{y^2}} \right) \\ + \left( {xy + 2xy - xy + 2xy} \right) \end{bmatrix} \\&= 3{x^2} - {y^2} +4xy\end{align}\]

vi) \(\left( {x{\rm{ }} + {\rm{ }}y} \right)\left( {{x^2}{\rm{ }} - {\rm{ }}xy{\rm{ }} + {\rm{ }}{y^2}} \right)\)

\[\begin{align}&=\begin{bmatrix} x\left( {{x^2} -xy + {y^2}} \right) \\ + y \left( {{x^2} - xy + {y^2}} \right)\end{bmatrix} \\&=\begin{bmatrix} {x^3} - {x^2}y + x{y^2} \\ + {x^2}y - x{y^2} + {y^3}\end{bmatrix} \\&=\begin{bmatrix} {x^3} + {y^3} + \\ \left( {x{y^2}- x{y^2}} \right) \\ + \left( {{x^2}y - {x^2}y} \right)\end{bmatrix} \\&= {\rm{ }}{x^3}{\rm{ }} + {y^3}\end{align}\]

vii)

\(\begin{bmatrix}\left( {1.5x{\rm{ }} - {\rm{ }}4y} \right)\left( {1.5x{\rm{ }} + {\rm{ }}4y{\rm{ }} + {\rm{ }}3} \right){\rm{ }}\\ - {\rm{ }}4.5x{\rm{ }} + {\rm{ }}12y \end{bmatrix}\)

\[\begin{align}&=\begin{bmatrix}1.5x \left( {1.5x + 4y + 3} \right) \\ - 4y \left( {1.5x + 4y + 3} \right) \\ - 4.5x + 12y\end{bmatrix} \\ &= \begin{bmatrix} 2.25 {x^2} + 6xy + 4.5x \\ - 6xy - 16{y^2} - 12y \\ - 4.5x + 12y \end{bmatrix} \\&=\begin{bmatrix} 2.25 {x^2} + \left( {6xy - 6xy} \right) \\ + \left( {4.5x - 4.5x} \right)- 16{y^2} \\ + \left( {12y - 12y} \right) \end{bmatrix} \\&= 2.25{x^2} - 16{y^2}\end{align}\]

viii) \(\left( {a{\rm{ }} + b + {\rm{ }}c} \right)\left( {a{\rm{ }} + b{\rm{ }} - {\rm{ }}c} \right)\)

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

  
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