Divide the given expression, x 2 - 8 x + 12 ÷ x - 6

We need to factorize the given expression, x 2 - 8 x + 12 ÷ x - 6 We need to split the middle term of the expression in numerator, x 2 - 8 x + 12 , Multiply the coefficient of the first term by the constant 1 × 12 = 12 . Now,find two factors of 12 whose sum equals the coefficient of the middle term, which is - 8 Thus, The factors of + 12 which sum to - 8 are - 6 and - 2 x 2 - 8 x + 12 = x 2 - 6 x + 2 x + 12 = x 2 - 6 x - 2 x + 12 = x x - 6 - 2 x - 6 = x - 2 x - 6 Hence,the factors of x 2 - 8 x + 12 are x - 2 x - 6 . Therefore, x 2 - 8 x + 12 x - 6 = x - 2 x - 6 x - 6 = x - 2 x 2 - 8 x + 12 ÷ x - 6 = x - 2

Divide the given expression, x 2 - 8 x + 12 ÷ x - 6

We need to factorize the given expression, x 2 - 8 x + 12 ÷ x - 6 We need to split the middle term of the expression in numerator, x 2 - 8 x + 12 , Multiply the coefficient of the first term by the constant 1 × 12 = 12 . Now,find two factors of 12 whose sum equals the coefficient of the middle term, which is - 8 Thus, The factors of + 12 which sum to - 8 are - 6 and - 2 x 2 - 8 x + 12 = x 2 - 6 x + 2 x + 12 = x 2 - 6 x - 2 x + 12 = x x - 6 - 2 x - 6 = x - 2 x - 6 Hence,the factors of x 2 - 8 x + 12 are x - 2 x - 6 . Therefore, x 2 - 8 x + 12 x - 6 = x - 2 x - 6 x - 6 = x - 2 x 2 - 8 x + 12 ÷ x - 6 = x - 2

Let log a m n = x ⇒ m n = a x ⇒ a x = m n . . . . . i Let log a m = b , log a n = c ⇒ m = a b , n = a c m n = a b a c = a b - c From equation i , ⇒ a x = a b - c ⇒ x = b - c ⇒ log a m n = log a m - log a n Hence, log m n = log m - log n .

Divide 15 y 4 + 16 y 3 + 10 3 y - 9 y 2 - 6 by 3 y - 2 . Write down the coefficients of the terms in the quotient.

Therefore, the quotient is 5 y 3 + 26 3 y 2 + 25 9 y + 80 27 . Now, we see, the following: Coefficient of y 3 = 5 , Coefficient of y 2 = 26 3 , Coefficient of y = 25 9 , Constant term = 80 27 . Therefore, the coefficients of the terms in the quotient are 5 , 26 3 , 25 9 , 80 27 .

Verify division algorithm i.e., Dividend = Divisor × Quotient + Remainder, in the following. Also, write the quotient and remainder: 14 x 2 + 13 x - 15 by 7 x - 4

2 x + 3 7 x - 4 14 x 2 + 13 x - 15 14 x 2 - 8 x - + 21 x - 15 21 x - 12 - + - 3 So, Quotient, = ( 2 x + 3 ) , reminder = - 3 . Now, verification: We have, Dividend = Divisor × Quotient + Remainder = 7 x - 4 × 2 x + 3 + - 3 = 7 x ( 2 x + 3 ) - 4 ( 2 x + 3 ) - 3 = 14 x 2 + 21 x - 8 x - 12 - 3 = 14 x 2 + 13 x - 15 = Dividend Hence, proved.

Divide by long division method: 3 x 5 + 5 x 4 - 8 x 3 + 5 x 2 - 8 x - 3 ÷ ( 3 x - 1 )

Given, 3 x 5 + 5 x 4 - 8 x 3 + 5 x 2 - 8 x - 3 ÷ ( 3 x - 1 ) . By using the division method, Hence, Q u o t i e n t = x 4 + 2 x 3 - 2 x 2 + x - 7 3 & R e m a i n d e r = - 16 3 .