Ex.14.1 Q1 Factorization - NCERT Maths Class 8

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Question

Find the common factors of the terms

(i) $$12x,\;\,36$$

(ii) $$2y,\;\,22xy$$

(iii) $$14pq,\;\,28{p^2}{q^2}$$

(iv) $$2x,\;\,3{x^2},\;\,4$$

(v) $$6abc,\;\,24a{b^2},\;\,12{a^2}b$$

(vi) $$16{x^3},\; - 4{x^2},\;32x$$

(vii) $$10pq,\;20qr,\;30rp$$

(viii) $$3{x^2}{y^3},\;\,10{x^3}{y^2},\;\,6{x^2}{y^2}z$$

Text Solution

What is known:

Terms.

What is unknown:

Common factors of given terms.

Reasoning:

First we will find factors of each terms then find out which factors are common in each term.

Steps:

\begin{align}(\rm{i})\quad 12x &= 2 \times 2 \times 3 \times x\\36 &= 2 \times 2 \times 3 \times 3\end{align}

The common factors are $$2, 2, 3.$$

And, $$~2\times 2\times 3=12$$

\begin{align}(\rm{ii})\quad 2y &= 2 \times y\\22xy &= 2 \times 11 \times x \times y\end{align}

The common factors are $$2, y.$$

And, $$2 \times y = 2y$$

\begin{align}(\rm{iii})\quad {\rm{14 }}pq &= {\rm{2}} \times {\rm{7}} \times p \times q\\{\rm{28}}{p^{\rm{2}}}{q^{\rm{2}}} &= {\rm{2}} \times {\rm{2}} \times {\rm{7}} \times p \times p \times q \times q\end{align}

The common factors are $$2, 7, p, q.$$

And, $$2 \times 7 \times p \times q = 14pq$$

\begin{align}(\rm{iv})\quad 2x &= {\rm{2}} \times x\\3x^2 &= 3 \times x \times x\\4 &= 2 \times 2\end{align}

The common factor is $$1.$$

\begin{align}(\rm{v}) \quad 6abc &= 2 \times 3 \times a \times b \times c\\24a{b^2} &= 2 \times 2 \times 2 \times 3 \times a \times b \times b\\12{a^2}b &= 2 \times 2 \times 3 \times a \times a \times b\end{align}

The common factors are $$2,\, 3,\, a,\, b.$$

And, $$2 \times 3 \times a \times b = 6ab$$

\begin{align}(\rm{vi})\quad16{x^3} &= 2 \times 2 \times 2 \times 2 \times x \times x \times x\\ - 4{x^2} &= - 1 \times 2 \times 2 \times x \times x\\32x &= 2 \times 2 \times 2 \times 2 \times 2 \times x\end{align}

The common factors are $$2,\, 2,\, x.$$

And,$$2\times 2\times x=4x$$

\begin{align}(\rm{vii})\quad 10pq &= 2 \times 5 \times p \times q\\20\,qr &= 2 \times 2 \times 5 \times q \times r\\30\,rp &= 2 \times 3 \times 5 \times r \times p\end{align}

The common factors are $$2, \;5.$$

And, $$2 \times 5 = 10$$

\begin{align}(\rm{viii})\quad 3{x^2}{y^3} &= 3 \times x \times x \times y \times y \times y\\ 10{x^3}{y^2} &= 2 \times 5 \times x \times x \times x \times y \times y\\6{x^2}{y^2}z &= 2 \times 3 \times x \times x \times y \times y \times z\end{align}

The common factors are $$x, \,x,\, y, \,y.$$

And, $$x \times x \times y \times y = {x^2}{y^2}$$

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