Q3 of 108 Page 46

If u(x) = (x – 1)2 and v(x) = (x2 – 1) the check the true of the relation LCM × HCF = u(x) × v(x).

u(x) = (x – 1)2

= (x – 1) (x – 1)


v(x) = (x2 – 1)


= (x + 1) (x – 1)


LCM of u(x) and v(x) = (x – 1)2 (x + 1)


HCF of u(x) and v(x) = (x – 1)


u(x) × v(x) = (x – 1) (x – 1) × (x2 – 1)


= (x2 – 2x + 1) × (x2 – 1)


= x4 – 2x3 + x2 - x2 + 2x – 1


= x4 – 2x3 + 2x – 1


HCF × LCM = (x – 1)2 (x + 1) × (x – 1)


= (x2 – 2x + 1) (x2 – 1)


= x4 – 2x3 + x2 - x2 + 2x – 1


= x4 – 2x3 + 2x – 1


So it is observed that HCF × LCM = u(x) × v(x).


Hence Proved.


More from this chapter

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1

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18(6x4 + x3 – x2) and 45(2x6 + 3x5 + x4)


 2

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)


 4

The product of two expressions is (x – 7) (x2 + 8x + 12). If the HCF of these expressions is (x + 6) then find their LCM.

 5

The HCF and LCM of two quadratic expressions are respectively (x – 5) and x3 – 19x – 30, then find both the expressions.