FIND GCD OF TWO POLYNOMIALS USING DIVISION METHOD

Find the GCD of the following pairs of polynomials using division algorithm

Problem 1 :

x³ - 9x² + 23x - 15 and 4x² - 16x + 12

Solution :

x³ - 9x² + 23x - 15 and 4x² - 16x + 12

Let f(x) = x³ - 9x² + 23x - 15 and

g(x) = 4x² - 16x + 12

g(x) = 4(x² - 4x + 3)

Here, degree of f(x) > degree of g(x)

By dividing f(x) by g(x), we get 0 as remainder. So, the required GCD is x - 5.

Problem 2 :

3x³ + 18x² + 33x + 18 and 3x² + 13x + 10

Solution :

Let f(x) = 3x³ + 18x² + 33x + 18 and g(x) = 3x² + 13x + 10

Here, degree f(x) > degree g(x)

By dividing f(x) by g(x) we are not getting zero. So, we have to do this long division once again.

Now we are taking 4/3 as common from the remainder. So that we are getting (4/3)(x+1)

So, the greatest common divisor is x + 1.

Problem 3 :

2x³ + 2x² + 2x + 2 and 6x³ + 12x² + 6x + 12

Solution :

Let f(x) = 2x³ + 2x² + 2x + 2 and g(x) = 6x³ + 12x² + 6x + 12

f(x) = 2(x³ + x² + x + 1)

g(x) = 6(x³ + 2x² + x + 2)

Since we are not getting zero, we have to do this long division once again

So, the required GCD is 2 (x² + 1).

Problem 4 :

x³ - 3x² + 4x - 12 and x⁴ + x³ + 4x² + 4x

Solution :

Let f(x) = x³ - 3x² + 4x - 12 and g(x) = x⁴ + x³ + 4x² + 4x

f(x) = x³ - 3x² + 4x - 12

g (x) = x(x³ + x² + 4x + 4)

Here, degree f(x) = degree g(x)

Since we are not getting zero, we have to do this long division once again

So, the required GCD is (x² + 4). 

Problem 5 :

x⁴ + 3x³ - x - 3 and x³ + x² - 5x + 3

Solution :

Let f(x) = x⁴ + 3x³ - x - 3 and g(x) = x³ + x² - 5x + 3

f(x) = x⁴ + 3x³ - x - 3

g (x) = x³ + x² - 5x + 3

Here, degree f(x) > degree g(x)

GCD of the polynomials is x² + 2x - 3

Problem 6 :

Find the GCD of 4 + 9x - 9x² and 9x² - 24x + 16

Solution :

Writing each polynomial in standard form, we get

Let f(x) = - 9x² + 9x + 4 and g(x) = 9x² - 24x + 16

Degree of polynomial f(x) = degree of polynomial g(x)

Since both polynomials are quadratic, by finding factoring and listing the common factors we will get GCD.

Let f(x) = - 9x² + 9x + 4

= -(9x² - 9x - 4)

= -(9x² - 12x + 3x - 4)

= -[3x(3x - 4) + 1(3x - 4)]

= -(3x + 1) (3x - 4) ------(1)

g(x) = 9x² - 24x + 16

= 9x² - 12x - 12x + 16

= 3x(3x - 4) - 4(3x - 4)

= (3x - 4)(3x - 4) ------(2)

From (1) and (2) common factors are 

Greatest common factor = (3x - 4)

Problem 7 :

Find the GCD of 8(x⁴ + x³ + x²) and 20(x³ - 1)

Solution :

Writing each polynomial in standard form, we get

Let f(x) = 8(x⁴ + x³ + x²) and g(x) = 20(x³ - 1)

Since both polynomials are quadratic, by finding factoring and listing the common factors we will get GCD.

Let f(x) = 8(x⁴ + x³ + x²)

f(x) = 8x²(x² + x + 1)

= 2³x²(x² + x + 1)-------(1)

g(x) = 20(x³ - 1)

g(x) = 2² ⋅ 5(x - 1)(x² + x + 1) -------(2)

Common factors are  2² and (x² + x + 1)

GCD = 4 (x² + x + 1)

Problem 8 :

Find the GCD of (3 + 13x - 30x²) and (25x² - 30x + 9)

a)  7x - 4    b)  5x - 3     c)  6x - 5     d) none

Solution :

Writing each polynomial in standard form, we get

Let f(x) = 3 + 13x - 30x² and g(x) = 25x² - 30x + 9

f(x) = -30 x² + 13x + 3

= -30 x² + 18x - 5x + 3

= -6x(5x - 3) - 1(5x - 3)

f(x) = (-6x - 1)(5x - 3) --------(1)

g(x) = 25x² - 30x + 9

= 25x² - 15x - 15x + 9

= 5x(5x - 3) - 3(5x - 3)

g(x) = (5x - 3)(5x - 3) --------(2)

The common factors are 5x - 3. So, the GCD is (5x - 3).

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