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

(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,

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

\({\rm{(iii) }}\;8\left( {{x^3}{y^2}{z^2} + {x^2}{y^3}{z^2} + {x^2}{y^2}{z^3}} \right) \div 4{x^2}{y^2}{z^2}\)

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,

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

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

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,

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