Find the highest common factor of the following expressions:
(i) a3b4, ab5, a2b8
(ii) 16x2y2, 48x4z
(iii) x2 – 7x + 12; x2 – 10x + 21 and x2 + 2x – 15
(iv) (x + 3)2 (x – 2) and (x + 3) (x – 2)2
(v) 24(6x4 – x3 – 2x2) and 20(6x6 + 3x5 + x4)
(i) Let u(x) = a3b4
Let v(x) = ab5
Let w(x) = a2b8
By comparing all the above equations, we get,
HCF = ab4 (Least power of a and b)
(ii) Let u(x) = 24 × x2 × y2
Let v(x) = 24 × 3 × x4 × z
By comparing all the above equations, we get,
HCF = 24 × x2
= 16 x2 (Least power of x, y and z and also common terms of u(x) and v(x))
(iii) Let u(x) = x2 – 7x + 12
= x2 – 4x – 3x + 12
= x(x – 4) – 3(x – 4)
= (x – 3) (x – 4)
Let v(x) = x2 – 10x + 21
= x2 – 7x – 3x + 21
= x(x – 7) – 3(x – 7)
= (x – 7) (x – 3)
Let w(x) = x2 + 2x – 15
= x2 + 5x – 3x – 15
= x(x + 5) – 3(x + 5)
= (x – 3) (x + 5)
By comparing all the above equations, we get,
HCF = (x + 3) [only common term from u(x), v(x) and w(x)].
(iv) Let u(x) = (x + 3)2 (x – 2)
Let v(x) = (x + 3) (x – 2)2
By comparing all the above equations, we get,
HCF = (x + 3) (x – 2) [Least power and common term from u(x) and v(x)].
(v) Let u(x) = (8 × 3) (6x4 – x3 – 2x2)
= (8 × 3) x2 (6x2 – x – 2)
= (8 × 3) x2 (6x2 – 4x + 3x – 2)
= (8 × 3) x2 (2x (3x – 2) + 1(3x – 2))
= (8 × 3) x2 (2x + 1) (3x – 2)
Let v(x) = 20(6x6 + 3x5 + x4)
= (4 × 5) x4(6x2 + 3x + 1)
= (4 × 5) x4(6x2 + 3x + 1)
By comparing all the above equations, we get,
HCF = 4 x2 (2x + 1) [Least power and common term from u(x) and v(x)].
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