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## Tamilnadu Samacheer Kalvi 10th Maths Solutions Chapter 3 Algebra Ex 3.3

Question 1.

Find the LCM and GCD for the following and verify that f(x) × g(x) = LCM × GCD.

(i) 21x^{2}y, 35xy^{2}

(ii) (x^{3} – 1)(x + 1), x^{3} + 1

(ii) (x^{3} – 1) (x + 1), (x^{3} – 1)

(iii) (x^{2}y + xy^{2}), (x^{2} + xy)

Solution:

(i) f(x) = 21x^{2}y = 3 × 7x^{2}y

g(x) = 35xy^{2} = 7 × 5xy^{2}

G.C.D. = 7xy

L.C.M. = 7 × 3 × 5 × x^{2}y^{2} = 105x^{2} × y^{2}

L.C.M × G.C.D = f(x) × g(x)

105x^{2}y^{2} × 7xy = 21x^{2}y × 35xy^{2}

735x^{3}y^{3} = 735x^{3}y^{3}

Hence verified.

(ii) (x^{3} – 1)(x + 1) = (x – 1)(x^{2} + x + 1)(x + 1)

x^{3} + 1 = (x + 1) (x^{2} – x + 1)

G.C.D = (x+ 1)

L.C.M = (x – 1)(x + 1)(x^{2} + x + 1)(x^{2} – x + 1)

∴ L.C.M. × G.C.D = f(x) × g(x)

(x – 1)(x + 1)(x^{2} + x + 1) (x^{2} – x + 1) = (x – 1)

(x^{2} + x + 1) × (x + 1) (x^{2} – x + 1)

(x^{3} – 1)(x + 1)(x^{3} + 1) = (x^{3} – 1)(x + 1)(x^{3} + 1)

∴ Hence verified.

(iii) f(x) = x^{2}y + xy^{2} = xy(x + y)

g(x) = x^{2} + xy = x(x + y)

L.C.M. = x y (x + y)

G.C.D. = x (x + y)

To verify:

L.C.M. × G.C.D. = xy(x + y) × (x + y)

= x^{2}y (x + y)^{2} ……….. (1)

f(x) × g (x) = (x^{2}y + xy^{2})(x^{2} + xy)

= x^{2}y (x + y)^{2} …………… (2)

∴ L.C.M. × G.C.D = f(x) × g{x).

Hence verified.

Question 2.

Find the LCM of each pair of the following polynomials

(i) a^{2} + 4a – 12, a^{2} – 5a + 6 whose GCD is a – 2

(ii) x^{4} – 27a^{3}x, (x – 3a)^{2} whose GCD is (x – 3a)

Solution:

(i) f(x) = a^{2} + 4a – 12 = (a + 6)(a – 2)

(ii) f(x) = x^{4} – 27a^{3}x = x(x^{3} – (3a)^{3})

g(x) = (x – 3a)^{2}

G.C.D = (x – 3a)

L.C.M. × G.C.D = f(x) × g(x)

L. C.M = \(\frac{x\left(x^{3}-(3 a)^{3}\right) \times(x-3 a)^{2}}{(x-3 a)}\)

L.C.M = x(x^{3} – (3a)^{3}) . (x – 3a)

= x(x – 3a)^{2} (x^{2} + 3ax + 9a^{2})

Question 3.

Find the GCD of each pair of the following polynomials

(i) 12(x^{4} – x^{3}), 8(x^{4} – 3x^{3} + 2x^{2}) whose LCM is 24x^{3} (x – 1)(x – 2)

(ii) (x^{3} + y^{3}), (x^{4} + x^{2}y^{2} + y^{4}) whose LCM is (x^{3} + y^{3}) (x^{2} + xy + y^{2})

Solution:

(i) f(x)= 12(x^{4} – x^{3})

g(x) = 8(x^{4} – 3x^{3} + 2x^{2})

L.C.M = 24x^{3} (x – 1)(x – 2)

(ii) (x^{3} + y^{3}), (x^{4} + x^{2}y^{2} + y^{4})

L.C.M. = (x^{3} + y^{3})(x^{2} + xy + y^{2})

Question 4.

Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table

Solution: