If α , β are the roots of quadratic equation a x 2 + b x + c = 0 , then find the value of α 2 + β 2 .

Since α and β are the roots of the quadratic equation f ( x ) = a x 2 + b x + c = 0 . The sum of roots for the quadratic equation, α + β = - b a The product of roots for the quadratic equation, α β = c a ( α + β ) 2 = α 2 + β 2 + 2 α β Now, substitute the value of the respective variables. ⇒ - b a 2 = α 2 + β 2 + 2 c a ⇒ b 2 a 2 = α 2 + β 2 + 2 c a ⇒ α 2 + β 2 = b 2 a 2 - 2 c a ⇒ α 2 + β 2 = b 2 - 2 a c a 2 Hence, b 2 - 2 a c a 2 is the required answer.

Divide 2 x 2 + 3 x + 1 by x + 2 .

Given: Dividend = 2 x 2 + 3 x + 1 Divisor = x + 2 Here are the steps required for Dividing by a Polynomial Containing More Than One Term (Long Division): Step 1 : Make sure the polynomial is written in descending order of their degrees. If any terms are missing, use a zero to fill in the missing term (this will help with the spacing). Step 2 : Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. Step 3 : Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol. Step 4 : Subtract and bring down the next term. Step 5 : Repeat Steps 2 , 3 and 4 . We stop the division process when either the remainder is zero or its degree is less than the degree of the divisor. 2 x - 1 x + 2 2 x 2 + 3 x + 1 2 x 2 + 4 x - x + 1 - x - 2 3 We know that, Dividend = Divisor × Quotient + Remainder ⇒ 2 x 2 + 3 x + 1 = x + 2 2 x - 1 + 3

Divide 3 x 3 + x 2 + 2 x + 5 by 1 + 2 x + x 2 .

Let p ( x ) = 3 x 3 + x 2 + 2 x + 5 and g ( x ) = 1 + 2 x + x 2 . Now, we need to divide p ( x ) by g ( x ) x 2 + 2 x + 1 3 x - 5 3 x 3 + x 2 + 2 x + 5 ( - ) 3 x 3 + ( - ) 6 x 2 + ( - ) 3 x - 5 x 2 - x + 5 - ( + ) 5 x 2 - ( + ) 10 x - ( + ) 5 9 x + 10