Find the zeros of the polynomial p x = x 2 - 15 x + 50

Given polynomial is p x = x 2 - 15 x + 50 To find the zeroes put p x = 0 Then, x 2 - 15 x + 50 = 0 x 2 - 5 x - 10 x + 50 = 0 x x - 5 - 10 x - 5 = 0 x - 10 x - 5 = 0 x = 10 and x = 5 Therefore, the zeros of the polynomial are x = 10 and x = 5 .

Find the quotient and the remainder when f x = 2 x 3 - 3 x 2 + 5 x - 7 is divided by g x = x - 3 using synthetic division.

Dividend in standard form 2 x 3 - 3 x 2 + 5 x - 7 Coefficient are 2 - 3 5 - 7 To find the zero of the divisor x - 3 , Put x - 3 = 0 x = 3 Now by synthetic division Steps of synthetic division (i) Write the coefficients of the dividend and use the zero of the linear factor in the divisor's place. (ii) Bring the first coefficient down and multiply it with the divisor. (iii) Write the product below the 2nd coefficient and add the column. (iv) Repeat until the last coefficient. The last number is taken as the remainder. (v) Take the coefficients and write the quotient. (vi) Note that the resultant polynomial is of one order less than the dividend polynomial. 3 2 - 3 5 - 7 0 6 9 42 2 3 14 35 Then, the quotient is 2 x 2 + 3 x + 14 , And the remainder is 35

If p and q are the roots of the equation x 2 - 3 x + 2 = 0 , find the value of 1 p - 1 q .

Given, p and q are the roots of the equation x 2 - 3 x + 2 = 0 . For a standard quadratic equation a x 2 + b x + c = 0 , the sum of the roots is given by - b a and product of the roots is given by c a . Here, a = 1 , b = - 3 and c = 2 . Sum of the roots = - - 3 1 ⇒ p + q = - - 3 1 ⇒ p + q = 3 _____________ 1 Product of the roots = 2 1 ⇒ p q = 2 1 ⇒ p q = 2 _______________ 2 Now, q - p 2 = q + p 2 - 4 p q ⇒ q - p 2 = 3 2 - 4 × 2 [From 1 and 2 ] ⇒ q - p = ± 9 - 8 ⇒ q - p = ± 1 ⇒ q - p p q = ± 1 p q ⇒ 1 p - 1 q = ± 1 2 ∵ p q = 2 Hence, 1 p - 1 q = ± 1 2