If f (x) and g(x) are two polynomials of same degree then the polynomial carrying the highest coefficient will be the dividend.
In case, if both have the same coefficient then compare the next least degree’s coefficient and proceed with the division.
If r (x) = 0 when f(x) is divided by g(x) then g(x) is called GCD of the polynomials.
Example 1 :
Find the GCD of the following polynomials.
x^{4} + 3x^{3} −x −3, x^{3} +x^{2} −5x + 3
Solution :
Let f(x) = x^{4} + 3x^{3} −x −3
g(x) = x^{3} +x^{2} −5x + 3
The degree of the polynomial f(x) is greater than g(x).
Note : If we have any missing term, we have to replace that place by 0.
The remainder is 3(x^{2} + 2x - 3), which is not equal to 0.So, we have to divide x^{3} +x^{2} −5x + 3 by x^{2} + 2x - 3 by leaving the remainder.
We get 0 as remainder by dividing the given polynomial by x^{2} + 2x - 3.
Hence the required GCD is x^{2} + 2x - 3.
Example 2 :
Find the GCD of the following polynomials.
x^{4} - 1, x^{3} - 11x^{2} + x - 11
Solution :
Let f(x) = x^{4} - 1
g(x) = x^{3} - 11x^{2} + x - 11
The degree of the polynomial f(x) is greater than g(x).
The remainder is 120(x^{2} + 1), which is not equal to 0.So, we have to divide x^{3} - 11x^{2} + x - 11 by 120(x^{2} + 1) by leaving the remainder.
Hence the G.C.D is x^{2} + 1
Example 3 :
Find the GCD of the following polynomials.
3x^{4} + 6x^{3} −12x^{2} −24x, 4x^{4} +14x^{3} + 8x^{2} −8x
Solution :
Let f(x) = 3x^{4} + 6x^{3} −12x^{2} −24x
f(x) = 3x(x^{3} + 2x^{2} - 4x - 8)
g(x) = 4x^{4} +14x^{3} + 8x^{2} −8x
g(x) = 2x (x^{3} + 7x^{2} + 4x - 4)
The remainder is 3(x^{2} + 4x + 4), which is not equal to 0.So, we have to divide x^{3} + 2x^{2} - 4x - 8 by x^{2} + 4x + 4 by leaving the remainder.
By dividing x^{2} + 4x + 4, we get 0 remainder. The term x is also common for both the polynomials.
Hence the G.C.D is x(x^{2} + 4x + 4).
Example 4 :
Find the GCD of the following polynomials.
3x^{3} + 3x^{2} + 3x + 3 , 6x^{3} +12x^{2} + 6x +12
Solution :
Let f(x) = 3x^{3} + 3x^{2} + 3x + 3
f(x) = 3(x^{3} + x^{2} + x + 1)
g(x) = 6x^{3} + 12x^{2} + 6x + 12
g(x) = 6(x^{3} + 2x^{2} + x + 2)
The remainder is (x^{2} + 0x + 1), which is not equal to 0.So, we have to divide x^{3} + x^{2} + x + 1 by x^{2} + 0x + 1 by leaving the remainder.
By dividing x^{2} + 1, we get 0 remainder. The constant 3 is also common for both the polynomials.
Hence the G.C.D is 3(x^{2} + 1).
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