Find the quadratic equation whose one root is 1 + 2 and the sum of its roots is 2 .

Let the given root be α = 1 + 2 Sum of roots = 2 So, another root β = 2 - 1 + 2 = 1 - 2 Product of roots = 1 + 2 1 - 2 = 1 - 2 2 = 1 - 2 = - 1 Equation is x 2 - x sum of roots + product of roots = 0 ⇒ x 2 - 2 x - 1 = 0 Thus, x 2 - 2 x - 1 = 0 is the required quadratic equation.

If the polynomial x 4 + 2 x 3 + 8 x 2 + 12 x + 18 is divided by x 2 + 5 and the remainder is in the form of p x + q , find the value of p and q .

Given: Dividend = x 4 + 2 x 3 + 8 x 2 + 12 x + 18 Divisor = x 2 + 5 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. x 2 + 2 x + 3 x 2 + 5 x 4 + 2 x 3 + 8 x 2 + 12 x + 18 x 4 + 5 x 2 2 x 3 + 3 x 2 + 12 x + 18 2 x 3 + 10 x 3 x 2 + 2 x + 18 3 x 2 + 15 2 x + 3 Remainder = 2 x + 3 , but it is given that remainder is of the form p x + q . ∴ p = 2 , q = 3 .

Find the quadratic equation whose one root is 1 + 2 and the sum of its roots is 2 .

Let the given root be α = 1 + 2 Sum of roots = 2 So, another root β = 2 - 1 + 2 = 1 - 2 Product of roots = 1 + 2 1 - 2 = 1 - 2 2 = 1 - 2 = - 1 Equation is x 2 - x sum of roots + product of roots = 0 ⇒ x 2 - 2 x - 1 = 0 Thus, x 2 - 2 x - 1 = 0 is the required quadratic equation.