EXERCISE 12.1 — Solutions
1. Find the common factors of the given terms.
(i) \(12x,\; 36\)
Prime factorise:
\(12x\) = \(2 \times 2 \times 3 \times x\);
36 = \(2 \times 2 \times 3 \times 3\).
Common prime factors: \(2 \times 2\) and \(3\).
So greatest common factor (Greatest Common Factor) is \(2 \times 2 \times 3 = 12.\)
All other common factors are : \(1,2,3,4,6,12\).
Answer: Greatest Common Factor = \(12\). Common factors: \(1,2,3,4,6,12\).
(ii) \(2y,\; 22xy\)
Prime factorise:
\(2y\) = \(2 \times y\);
\(22xy\) = \(2 \times 11 \times x \times y\).
Common prime factors: \(2 \times y\).
So Greatest Common Factor = \(2y\).
All other common factors are : \(1,2,y,2y\).
Answer: Greatest Common Factor = \(2y\). Common factors: \(1,2,y,2y\).
(iii) \(14pq,\; 28p^{2}q\)
Prime factorise:
\(14pq\) = \(2 \times 7 \times p \times q\);
\(28p^{2}q\) = \(2 \times 2 \times 7 \times p \times p \times q\).
Common prime factors: \(2 \times 7 \times p \times q\).
So Greatest Common Factor = \(14pq\).
Answer: Greatest Common Factor = \(14pq\).
(iv) \(2x,\; 3x^{2},\; 4\)
Prime factorise:
\(2x\) = \(2 \times x\);
\(3x^{2}\) = \(3 \times x \times x\);
\(4\) = \(2 \times 2\).
No variable is common in all terms. Numeric gcd of \(2,3,4\) is 1.
Answer: Greatest Common Factor = \(1\). Common factor = 1 only.
(v) \(6abc,\; 24ab^{2},\; 12a^{2}b\)
Prime factorise:
\(6abc\) = \(2 \times 3 \times a \times b \times c\);
\(24ab^{2}\) = \(2 \times 2 \times 2 \times 3 \times a \times b \times b\);
\(12a^{2}b\) = \(2 \times 2 \times 3 \times a \times a \times b\).
Common factors: \(2 \times 3 \times a \times b = 6ab\).
Answer: Greatest Common Factor = \(6ab\).
(vi) \(16x^{3},\; -4x^{2},\; 32x\)
Prime factorise:
\(16x^{3}\) = \(2 \times 2 \times 2 \times 2 \times x \times x \times x\);
\(-4x^{2}\) = \(-1 \times 2 \times 2 \times x \times x\);
\(32x\) = \(2 \times 2 \times 2 \times 2 \times 2 \times x\).
Common factors: \(2 \times 2 \times x = 4x\).
Answer: Greatest Common Factor = \(4x\).
(vii) \(10pq,\; 20qr,\; 30rp\)
Prime factorise:
\(10pq\) = \(2 \times 5 \times p \times q\);
\(20qr\) = \(2 \times 2 \times 5 \times q \times r\);
\(30rp\) = \(2 \times 3 \times 5 \times r \times p\).
No variable common in all three. Numeric gcd = \(2 \times 5 = 10\).
Answer: Greatest Common Factor = \(10\).
(viii) \(3x^{2}y^{3},\; 10x^{3}y^{2},\; 6x^{2}y^{2}z\)
Prime factorise:
\(3x^{2}y^{3}\) = \(3 \times x \times x \times y \times y \times y\);
\(10x^{3}y^{2}\) = \(2 \times 5 \times x \times x \times x \times y \times y\);
\(6x^{2}y^{2}z\) = \(2 \times 3 \times x \times x \times y \times y \times z\).
Common variables: \(x^{2}\) and \(y^{2}\). Numeric gcd = 1.
Answer: Greatest Common Factor = \(x^{2}y^{2}\).
2. Factorise the following expressions.
(i) \(7x - 42\)
Common factor is \(7\).
Divide each term: \(7x ÷ 7 = x,\; -42 ÷ 7 = -6\).
Answer: \(7x - 42 = 7(x - 6)\).
(ii) \(6p - 12q\)
Common factor is \(6\).
Divide each term: \(6p ÷ 6 = p,\; -12q ÷ 6 = -2q\).
Answer: \(6p - 12q = 6(p - 2q)\).
(iii) \(7a^{2} + 14a\)
Common factor is \(7a\).
Divide each term: \(7a^{2} ÷ 7a = a,\; 14a ÷ 7a = 2\).
Answer: \(7a^{2} + 14a = 7a(a + 2)\).
(iv) \(-16z + 20z^{3}\)
Common factor is \(4z\).
Divide each term: \(-16z ÷ 4z = -4,\; 20z^{3} ÷ 4z = 5z^{2}\).
Answer: \(-16z + 20z^{3} = 4z(5z^{2} - 4)\).
(v) \(20l^{2}m + 30alm\)
Common factor is \(10lm\).
Divide each term: \(20l^{2}m ÷ 10lm = 2l,\; 30alm ÷ 10lm = 3a\).
Answer: \(20l^{2}m + 30alm = 10lm(2l + 3a)\).
(vi) \(5x^{2}y - 15xy^{2}\)
Common factor is \(5xy\).
Divide each term: \(5x^{2}y ÷ 5xy = x,\; -15xy^{2} ÷ 5xy = -3y\).
Answer: \(5x^{2}y - 15xy^{2} = 5xy(x - 3y)\).
(vii) \(10a^{2} - 15b^{2} + 20c^{2}\)
Common factor is \(5\).
Divide each term: \(10a^{2} ÷ 5 = 2a^{2},\; -15b^{2} ÷ 5 = -3b^{2},\; 20c^{2} ÷ 5 = 4c^{2}\).
Answer: \(10a^{2} - 15b^{2} + 20c^{2} = 5(2a^{2} - 3b^{2} + 4c^{2})\).
(viii) \(-4a^{2} + 4ab - 4ca\)
Common factor is \(4a\).
Divide each term: \(-4a^{2} ÷ 4a = -a,\; 4ab ÷ 4a = b,\; -4ca ÷ 4a = -c\).
Answer: \(-4a^{2} + 4ab - 4ca = 4a(-a + b - c)\).
(ix) \(x^{2}yz + xy^{2}z + xyz^{2}\)
Common factor is \(xyz\).
Divide each term: \(x^{2}yz ÷ xyz = x,\; xy^{2}z ÷ xyz = y,\; xyz^{2} ÷ xyz = z\).
Answer: \(x^{2}yz + xy^{2}z + xyz^{2} = xyz(x + y + z)\).
(x) \(ax^{2}y + bxy^{2} + cxyz\)
Common factor is \(xy\).
Divide each term: \(ax^{2}y ÷ xy = ax,\; bxy^{2} ÷ xy = by,\; cxyz ÷ xy = cz\).
Answer: \(ax^{2}y + bxy^{2} + cxyz = xy(ax + by + cz)\).
3. Factorise.
(i) \(x^{2} + xy + 8x + 8y\)
Group terms: \((x^{2} + xy) + (8x + 8y)\).
Factor each: \(x(x + y) + 8(x + y)\).
Take common factor \((x + y)\).
Answer: \((x + y)(x + 8)\).
(ii) \(15xy - 6x + 5y - 2\)
Group terms: \((15xy - 6x) + (5y - 2)\).
Factor: \(3x(5y - 2) + 1(5y - 2)\).
Take common factor \((5y - 2)\).
Answer: \((5y - 2)(3x + 1)\).
(iii) \(ax + bx - ay - by\)
Group terms: \((ax + bx) - (ay + by)\).
Factor: \(x(a + b) - y(a + b)\).
Take common factor \((a + b)\).
Answer: \((a + b)(x - y)\).
(iv) \(15pq + 15 + 9q + 25p\)
Group terms: \((15pq + 9q) + (25p + 15)\).
Factor: \(3q(5p + 3) + 5(5p + 3)\).
Take common factor \((5p + 3)\).
Answer: \((5p + 3)(3q + 5)\).
(v) \(z - 7 + 7xy - xyz\)
Group terms: \((z - xyz) + (7xy - 7)\).
Factor: \(z(1 - xy) + 7(xy - 1)\).
Note: \(7(xy - 1) = -7(1 - xy)\).
So expression = \((1 - xy)(z - 7)\).
Answer: \((z - 7)(1 - xy)\).
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