Ex.14.3 Q1 Factorization - NCERT Maths Class 8

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Question

 Carry out the following divisions.

(i)\(\begin{align}\quad 28{x^4} \div 56x\end{align}\)

(ii)\(\begin{align}\quad - 36{y^3} \div 9{y^2}\end{align}\)

(iii)\(\begin{align}\quad 66p{q^2}{r^3} \div 11q{r^2}\end{align}\)

(iv)\(\begin{align}\quad 34{x^3}{y^3}{z^3} \div 51x{y^2}{z^3}\end{align}\)

(v)\(\begin{align}\quad 12{a^8}{b^8} \div ( { - 6{a^6}{b^4}} )\end{align}\)

Text Solution

(i)\(\,28{x^4} \div 56x\)

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor of \(28{x}^4\) and \(56{x}\) then cancel out common factor of\(28{x}^4\) and \(56{x}\).

Steps:

(\(28{x^4}\)can be written as\(2 \times 2 \times 7 \times x \times x \times x \times x\)) and (\(56x\) can be written as\(2 \times 2 \times 2 \times 7 \times x\))

Then,

\[\begin{align}28{x^4} \div 56x &= \frac{{2 \times 2 \times 7 \times x \times x \times x \times x}}{{2 \times 2 \times 2 \times 7 \times x}}\\ &= \frac{{{x^3}}}{2}\\ &= \frac{1}{2}{x^3}\end{align}\]

(ii)\(\,- 36{y^3} \div 9{y^2}\)

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor of \(-36{y^3}\) and \(9{y^2}\) then cancel out common factor of \(-36{y^3}\) and \(9{y^2}\)

(\(- 36{y^3}\)can be written as\(- 2 \times 2 \times 3 \times 3 \times y \times y \times y\)) and (\(9{y^2}\)can be written as\(3 \times 3 \times y \times y\))

Then,

\[\begin{align}- 36{y^3} \div 9{y^2} &= \frac{{ - 2 \times 2 \times 3 \times 3 \times y \times y \times y}}{{3 \times 3 \times y \times y}}\\ &= - 4y\end{align}\]

(iii)\(\;66p{q^2}{r^3} \div 11q{r^2}\)

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor of \(66p{q^2}\) and \(11p{r^2}\) then cancel out common factor of \(66p{q^2}\) and \(11q{r^2}\)

(\(66p{q^2}{r^3}\) can be written as \(2 \times 3 \times 11 \times p \times q \times q \times r \times r \times r\)) and (\(11q{r^2}\)can be written as \(11 \times q \times r \times r\))

Then,

\[\begin{align} 66p{q^2}{r^3} \div 11q{r^2} &= \frac{{2 \times 3 \times 11 \times p \times q \times q \times r \times r \times r}}{{11 \times q \times r \times r}}\\&= 6pqr\end{align}\]

(iv)\(\,34{x^3}{y^3}{z^3} \div 51x{y^2}{z^3}\)

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor of \(34{x^3}{y^3}{z^3}\) and \(51x{y^2}{z^3}\) then cancel out common factor of \(34{x^3}{y^3}{z^3}\) and \(51x{y^2}{z^3}\)

(\(34{x^3}{y^3}{z^3}\) can be written as\(2 \times 17 \times x \times x \times x \times y \times y \times y \times z \times z \times z\)) and (\(51x{y^2}{z^3}\) can be written as \(3 \times 17 \times x \times y \times y \times z \times z \times z\))

Then,

\[\begin{align}34{x^3}{y^3}{z^3} \div 51x{y^2}{z^3} &= \frac{{2 \times 17 \times x \times x \times x \times y \times y \times y \times z \times z \times z}}{{3 \times 17 \times x \times y \times y \times z \times z \times z}}\\&= \frac{2}{3}{x^2}y\end{align}\]

(v)\(\;12{a^8}{b^8} \div \left( { - 6{a^6}{b^4}} \right)\)

What is known?

Algebraic expression.

What is unknown?

Division of the algebraic expression.

Reasoning:

Find out factor of \(12{a^8}{b^8}\) and \(- 6{a^6}{b^4}\) then cancel out common factor of \(- 6{a^6}{b^4}\) and \(51x{y^2}{z^3}\)

(\(12{a^8}{b^8}\)can be written as\(2 \times 2 \times 3 \times {a^8} \times {b^8}\)) and (\(- 6{a^6}{b^4}\)can be written as\(- 2 \times 3 \times {a^6} \times {b^4}\))

Then,

\[\begin{align}12{a^8}{b^8} \div \left( { - 6{a^6}{b^4}} \right) &= \frac{{2 \times 2 \times 3 \times {a^8} \times {b^8}}}{{ - 2 \times 3 \times {a^6} \times {b^4}}}\\&= - 2{a^2}{b^4}\end{align}\]