Question

Factorise the following expressions 

1.  \(7x - 42\) 
2.  \(6p - 12q\) 
3.  \(7a ^2 + 14a\) 
4.  \(-16z + 20z ^3\) 
5.  \(20l^ 2m + 30\, alm\) 
6.  \(5x^ 2 y - 15xy^2\) 
7.  \(10a^ 2 - 15b^ 2 + 20c^ 2 \)
8.  \(-4a^ 2 + 4ab - 4 ca\) 
9.  \(x ^2 yz + xy^2 z + xyz^2\)
10.  \(ax^2 y + bxy^2 + cxyz\)

Answer 1

(i) \(7x = 7 \times x\)
\(42 = 2 \times 3 \times 7\)
The common factor is \(7\).
∴ \(7x - 42 = (7 \times x) - (2 \times 3 \times 7) = 7 (x - 6)\)

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(ii) \(6p = 2 \times 3 \times p\)
\(12q = 2 \times 2 \times 3 \times q\)
The common factors are \(2\) and \(3\).
∴ \(6p - 12q = (2 \times 3 \times p) - (2 \times 2 \times 3 \times q)\)
= \(2 \times 3 [p - (2 \times q)]\)
= \(6 (p - 2q)\)

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(iii) \(7a^2= 7 \times a \times a\)
\(14a = 2 \times 7 \times a\)
The common factors are \(7\) and \(a\).
∴ \(7a^2+ 14a = (7 \times a \times a) + (2 \times 7 \times a)\)
= \(7 \times a [a + 2] = 7a (a + 2)\)

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(iv) \(16z = 2 \times 2 \times 2 \times 2 \times z \\\)
\(20z^3= 2 \times 2 \times 5 \times z \times z \times z\)
The common factors are \(2\), \(2\), and \(z\).
∴ \(-16z + 20z^3 = - (2 \times 2 \times 2 \times 2 \times z) + (2 \times 2 \times 5 \times z \times z \times z)\)
= \((2 \times 2 \times z) [- (2 \times 2) + (5 \times z \times z)]\)
= \(4z (- 4 + 5z^2)\)

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(v) \(20l^2m = 2 \times 2 \times 5 \times l \times l \times m\)
\(30alm = 2 \times 3 \times 5 \times a \times l \times m\)
The common factors are \(2, 5\), \(l\), and \(m\).
∴ \(20l^2m + 30alm = (2 \times 2 \times 5 \times l \times l \times m) + (2 \times 3 \times 5 \times a \times l \times m)\)
= \((2 \times 5 \times l \times m) [(2 \times l) + (3 \times a)]\)
= \(10lm\; (2l + 3a)\)

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(vi) \(5x^2y = 5 \times x \times x \times y\)
\(15xy^2= 3 \times 5 \times x \times y \times y\)
The common factors are \(5\), \(x\), and \(y\).
∴ \(5x^2y - 15xy^2\)
= \((5 \times x \times x \times y) - (3 \times 5 \times x \times y \times y)\)
= \(5 \times x \times y [x - (3 \times y)]\)
= \(5xy (x - 3y)\)

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(vii) \(10a^2= 2 \times 5 \times a \times a\)
\(15b^2= 3 \times 5 \times b \times b\)
\(20c^2= 2 \times 2 \times 5 \times c \times c\)
The common factor is \(5\).
\(10a^2- 15b^2+ 20c^2\)
= \((2 \times 5 \times a \times a) - (3 \times 5 \times b \times b) + (2 \times 2 \times 5 \times c \times c)\)
= \(5 [(2 \times a \times a) - (3 \times b \times b) + (2 \times 2 \times c \times c)]\)
= \(5 (2a^2- 3b^2+ 4c^2)\)

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(viii) \(4a^2= 2 \times 2 \times a \times a\)
\(4ab = 2 \times 2 \times a \times b\)
\(4ca = 2 \times 2 \times c \times a\)
The common factors are \(2, 2,\) and \(a\).
∴ \(-4a^2+ 4ab - 4ca = - (2 \times 2 \times a \times a) + (2 \times 2 \times a \times b) - (2 \times 2 \times c \times a)\)
=\( 2 \times 2 \times a [- (a) + b - c]\)
= \(4a (-a + b - c)\)

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(ix) \(x^2yz = x \times x \times y \times z\)
\(xy^2z = x \times y \times y \times z\)
\(xyz^2 = x \times y \times z \times z\)
The common factors are \(x, y\), and \(z\).
∴ \(x^2yz + xy^2z + xyz^2\)
= \((x \times x \times y \times z) + (x \times y \times y \times z) + (x \times y \times z \times z)\)
= \(x \times y \times z [x + y + z]\)
= \(xyz (x + y + z)\)

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(x) \(ax^2y = a \times x \times x \times y\)
\(bxy^2 = b \times x \times y \times y\)
\(cxyz = c \times x \times y \times z\)
The common factors are \(x\) and \(y\).