(x³-1) (x+1), (x³+1) find gcd
Answers
answer :
Greatest Common Divisor (GCD) :
Greatest Common Divisor (GCD) or (HCF) of two or more algebraic expressions is the expression of highest degree which divides each of them without remainder.
Method to find the GCD or HCF of polynomials :
1.Find the HCF of the common numerical Coefficient if any.
2. Factorise each of the given expressions and take the factors common to all of them.
3. The product of the number and factors obtained in steps 1 and 2 is the required as GCD or HCF.
SOLUTION :
vi) 2 x² - x - 1 , 4 x² + 8 x + 3
•2 x² - x - 1 = 2x² - 2x +1x -1
[By Middle term splitting]
2x(x -1) +1(x-1)
(2 x + 1) (x - 1)
•4 x² + 8 x + 3 = 4x² +6x +2x +3
[By Middle term splitting]
2x(2x +3) +1(2x+3)
(2x + 3) (2x + 1)
The factor (2 x + 1) is common to all the given expressions.
Hence GCD = (2 x + 1)
vii)
•x² - x - 2 = x² -2x +1x -2
[By Middle term splitting]
x(x -2) +1(x -2)
(x + 1) (x - 2)
•x² + x - 6 = x² + 3x -2x -6
[By Middle term splitting]
x(x +3) -2(x +3)
(x + 3) (x - 2)
•3 x² - 13 x + 14 = 3x² -7x - 6x +14
[By Middle term splitting]
x(3x -7) - 2(3x -7)
(x - 2) (3 x - 7)
The factor (x - 2) is common to all the given expressions.
Hence GCD =(x - 2)
viii)
x³ - x² + x - 1 , x⁴ - 1
x³ - x² + x - 1
Factors of constant term 1 is ±1
Put x = 1 , We get zero so (x-1) is a factor of this Polynomial.
Divide x³ - x² + x - 1 /(x -1) = x² +1
•x³ - x² + x - 1 = (x - 1) (x² + 1)
•x⁴ - 1 = (x²)² - (1²)²
[(a² - b² ) = (a +b (a-b)]
= (x² + 1) (x² - 1)
= (x² + 1) (x + 1) (x - 1)
The factor (x - 1) (x² + 1) is common to all the given expressions.
Hence GCD =(x - 1) (x² + 1)
ix)
24 (6x⁴ - x³ - 2x²) , 20 (2 x⁶ + 3 x⁵ + x⁴)
•24 (6x⁴ - x³ - 2x²) = 24 x² (6x² - x - 2)
= 24 x² (6x² - 4x +3x -2)
[By Middle term splitting]
= 24 x² (2x(3x -2) + 1(3x -2)
= 2 x 3 x 2 x 2 x x² (3x - 2) (2x + 1)
•20 (2x⁶ + 3 x⁵ + x⁴) = 20 x⁴ (2 x² + 3 x + 1)
= 20 x⁴ (2x² +2x +1x +1)
[By Middle term splitting]
= 20 x⁴ ( 2x (x +1) +1(x +1)
= 5 x 2 x 2 x x² x x² (x + 1) (2x + 1)
The factor 2 x 2 x x² x (2x + 1) is common to all the given expressions.
Hence GCD = 4x² (2x + 1)
x)
(a - 1)⁵ (a + 3)² , (a - 2)² (a - 1)³ (a + 3)⁴
•(a - 1)⁵ (a + 3)² = (a - 1)³ (a - 1)² (a + 3)²
•(a - 2)² (a - 1)³ (a + 3)⁴ = (a - 2)² (a - 1)³ (a + 3)² (a + 3)²
The factor (a - 1)³ (a + 3)² is common to all the given expressions.
Hence GCD = (a - 1)³ (a + 3)²
HOPE THIS WILL HELP YOU.