Samacheer Kalvi 10th Maths Solutions Chapter 3 Algebra Ex 3.3

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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) 21x²y, 35xy²
(ii) (x³ – 1)(x + 1), x³ + 1
(ii) (x³ – 1) (x + 1), (x³ – 1)
(iii) (x²y + xy²), (x² + xy)
Solution:
(i) f(x) = 21x²y = 3 × 7x²y
g(x) = 35xy² = 7 × 5xy²
G.C.D. = 7xy
L.C.M. = 7 × 3 × 5 × x²y² = 105x² × y²
L.C.M × G.C.D = f(x) × g(x)
105x²y² × 7xy = 21x²y × 35xy²
735x³y³ = 735x³y³
Hence verified.

(ii) (x³ – 1)(x + 1) = (x – 1)(x² + x + 1)(x + 1)
x³ + 1 = (x + 1) (x² – x + 1)
G.C.D = (x+ 1)
L.C.M = (x – 1)(x + 1)(x² + x + 1)(x² – x + 1)
∴ L.C.M. × G.C.D = f(x) × g(x)
(x – 1)(x + 1)(x² + x + 1) (x² – x + 1) = (x – 1)
(x² + x + 1) × (x + 1) (x² – x + 1)
(x³ – 1)(x + 1)(x³ + 1) = (x³ – 1)(x + 1)(x³ + 1)
∴ Hence verified.

(iii) f(x) = x²y + xy² = xy(x + y)
g(x) = x² + 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)
= x²y (x + y)² ……….. (1)
f(x) × g (x) = (x²y + xy²)(x² + xy)
= x²y (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) a² + 4a – 12, a² – 5a + 6 whose GCD is a – 2
(ii) x⁴ – 27a³x, (x – 3a)² whose GCD is (x – 3a)
Solution:
(i) f(x) = a² + 4a – 12 = (a + 6)(a – 2)

(ii) f(x) = x⁴ – 27a³x = x(x³ – (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(x³ – (3a)³) . (x – 3a)
= x(x – 3a)² (x² + 3ax + 9a²)

Question 3.
Find the GCD of each pair of the following polynomials
(i) 12(x⁴ – x³), 8(x⁴ – 3x³ + 2x²) whose LCM is 24x³ (x – 1)(x – 2)
(ii) (x³ + y³), (x⁴ + x²y² + y⁴) whose LCM is (x³ + y³) (x² + xy + y²)
Solution:
(i) f(x)= 12(x⁴ – x³)
g(x) = 8(x⁴ – 3x³ + 2x²)
L.C.M = 24x³ (x – 1)(x – 2)

(ii) (x³ + y³), (x⁴ + x²y² + y⁴)
L.C.M. = (x³ + y³)(x² + xy + y²)

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: