Question

Hey friends

Check whether the first polynomial is a factor of the second polynomial by applying Euclid's division algorithm for each of the following .

$x^{2}-5x+6\:\:\:\:\:\:; 10x^{4}+8x^{2}+6x+3$

Hint:Answer given in book is" Yes"

So please solve it

NO SPAM OR COPYING FROM THE INTERNET

Answer 1

Step-by-step explanation:

Given Equation is x² - 5x + 6.

Given Equation is 10x⁴ + 8x² + 6x + 3.

Euclid's division algorithm:

Dividend = Quotient * Divisor + Remainder

10x⁴ + 8x² + 6x + 3 = g(x) * q(x) + r(x)

10x⁴ + 8x² + 6x + 3 = (x² - 5x + 6)(ax² + bx + c) + (px + q)

10x⁴ + 0x³ +  8x² + 6x + 3 = ax⁴ + (b - 5a)x³ + (c - 5b + 6a)x² + (p - 5c + 6b) x + 6c + q

Equating the coefficient of powers of x, we get

a = 10, (b - 5a) = 0, c - 5b + 6a = 8, p - 5c + 6b = 6, 6c + q = 3

(i)

b - 5a = 0

b - 5(10) = 0

b = 50.

(ii)

c - 5b + 6a = 8

c - 5(50) + 6(10) = 8

c - 250 + 60 = 8

c = 198

(iii)

6c + q = 3

6(198) + q = 3

1188 + q = 3

q = -1185

(iv)

p - 5c + 6b = 6

p - 5(198) + 6(50) = 6

p - 990 + 300 = 6

p = 696.

So,Quotient = 10x² + 50x + 198,Remainder = 696x - 1185.

Here,the remainder is not equal to 0.

Therefore g(x) is not a factor of p(x).

-------------------------------------------------------------------------------------------

Long-division method:

x² - 5x + 6) 10x⁴ + 0x³ + 8x² + 6x + 3 ( 10x² + 50x + 198

                  10x⁴ - 50x³ + 60x²

                  -----------------------------------

                           50x³ - 52x² + 6x + 3

                          50x² - 250x² + 300x + 3

                  ------------------------------------------

                                      198x² - 294x + 3

                                      198x² - 990x + 1188

                    -------------------------------------------

                                                     696x - 1185

Here, remainder is not equal to 0.

Therefore, g(x) is not a factor of f(x).

Hope this helps!