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:

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

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, \(2 \times y = 2y\)

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

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 \(1.\)

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

The common factors are \(2,\, 3,\, a,\, b.\)

And, \(2 \times 3 \times a \times b = 6ab\)

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

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, \(2 \times 5 = 10\)

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

The common factors are  \(x, \,x,\, y, \,y.\)

And, \(x \times x \times y \times y = {x^2}{y^2}\)

  
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