# Ex.14.3 Q2 Factorization - NCERT Maths Class 8

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## Question

Divide the given polynomial by the given monomial.

(i) \begin{align} \quad \left( {5{x^2} - 6x} \right) \div 3x\end{align}

(ii) \begin{align} \quad \left( {3{y^8} - 4{y^6} + 5{y^4}} \right) \div {y^4}\end{align}

(iii) \begin{align} \begin{Bmatrix} 8\left( {{x^3}{y^2}{z^2} + {x^2}{y^3}{z^2} + {x^2}{y^2}{z^3}} \right) \\ \div 4{x^2}{y^2}{z^2} \end{Bmatrix} \end{align}

(iv) \begin{align} \quad \left( {{x^3} + 2{x^2} + 3x} \right) \div 2x\end{align}

(v) \begin{align} \quad \left( {{p^3}{q^6} - {p^6}{q^3}} \right) \div {p^3}{q^3}\end{align}

## Text Solution

$${\rm{(i) }} \;\left( {5{x^2} - 6x} \right) \div 3x$$

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor $$({5{x^2} - 6x})$$ and $$3{x}$$ then cancel out common factors of $$({5{x^2} - 6x})$$ and $$3{x}$$.

Steps:

$$5{x^2} - 6x$$ can be written as $$x(5x - 6)$$

Then,

\begin{align} \left( {5{x^2} - 6x} \right) \div 3x &= \frac{{x(5x - 6)}}{{3x}}\\&= \frac{1}{3}(5x - 6)\end{align}

$${\rm{(ii) }}\;( {3{y^8} - 4{y^6} + 5{y^4}}) \div {y^4}$$

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor $$( {3{y^8} - 4{y^6} + 5{y^4}})$$ and $${y^4}$$ then cancel out common factors of $$( {3{y^8} - 4{y^6} + 5{y^4}})$$ and $${y^4}$$.

Steps:

$$3{y^8} - 4{y^6} + 5{y^4}$$ can be written as $${y^4}\left( {3{y^4} - 4{y^2} + 5} \right)$$

Then,

\begin{align} & \left( {3{y^8} - 4{y^6} + 5{y^4}} \right) \div {y^4} \\ & = \frac{{{y^4}\left( {3{y^4} - 4{y^2} + 5} \right)}}{{{y^4}}}\\&= 3{y^4} - 4{y^2} + 5\end{align}

(iii)

\ \begin{align} \begin{Bmatrix} 8\left( {{x^3}{y^2}{z^2} + {x^2}{y^3}{z^2} + {x^2}{y^2}{z^3}} \right) \\ \div 4{x^2}{y^2}{z^2} \end{Bmatrix} \end{align}

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor $$8\left( {{x^3}{y^2}{z^2} + {x^2}{y^3}{z^2} + {x^2}{y^2}{z^3}} \right)$$ and $$4{x^2}{y^2}{z^2}$$ then cancel out common factors of $$8\left( {{x^3}{y^2}{z^2} + {x^2}{y^3}{z^2} + {x^2}{y^2}{z^3}} \right)$$ and $$4{x^2}{y^2}{z^2}$$.

Steps:

$$8\left( {{x^3}{y^2}{z^2} + {x^2}{y^3}{z^2} + {x^2}{y^2}{z^3}} \right)$$ can be written as $$8{x^2}{y^2}{z^2}(x + y + z)$$

Then,

\begin{align}& \begin{Bmatrix} 8\left( {{x^3}{y^2}{z^2} + {x^2}{y^3}{z^2} + {x^2}{y^2}{z^3}} \right) \\ \div 4{x^2}{y^2}{z^2}\end{Bmatrix} \\ \\ &= \frac{{8{x^2}{y^2}{z^2}(x + y + z)}}{{4{x^2}{y^2}{z^2}}}\\&= 2(x + y + z)\end{align}

$${\rm{(iv)}} \,({{x^3} + 2{x^2} + 3x}) \div 2x$$

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor $$({{x^3} + 2{x^2} + 3x})$$ and $$2{x}$$ then cancel out common factors of $$({{x^3} + 2{x^2} + 3x})$$ and $$2{x}$$.

Steps:

$${x^3} + 2{x^2} + 3x$$ can be written as $$x\left( {{x^2} + 2x + 3} \right)$$

Then,

\begin{align} & \left( {{x^3} + 2{x^2} + 3x} \right) \div 2x\\ \\ &= \frac{{x\left( {{x^2} + 2x + 3} \right)}}{{2x}}\\&= \frac{1}{2}\left( {{x^2} + 2x + 3} \right)\end{align}

$$({\rm{v}})\;( {{p^3}{q^6} - {p^6}{q^3}}) \div {p^3}{q^3}$$

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor $$({{p^3}{q^6} - {p^6}{q^3}})$$ and $${p^3}{q^3}$$ then cancel out common factors of $$( {{p^3}{q^6} - {p^6}{q^3}})$$ and $${p^3}{q^3}$$.

Steps:

$${p^3}{q^6} - {p^6}{q^3}$$ can be written as $${p^3}{q^3}\left( {{q^3} - {p^3}} \right)$$

Then,

\begin{align} & \left( {{p^3}{q^6} - {p^6}{q^3}} \right) \div {p^3}{q^3} \\ \\ &= \frac{{{p^3}{q^3}\left( {{q^3} - {p^3}} \right)}}{{{p^3}{q^3}}}\\&= {q^3} - {p^3}\end{align}

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