Question

If x-3 and x - 1/3 are both factor of ax^2+5x,b, show that a-b
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Answer 1

Correct question : x - 3 and x - ⅓ are both the factors of ax² + 5x + b.

To prove : To show that a = b.

$\qquad$_______________

The zeroes of the polynomial ax² + 5x + b are :

$\qquad$ ★ x - 3 = 0

$\dashrightarrow~$ x = 3

$\qquad$ ★ x - ⅓ = 0

$\dashrightarrow~$ x = ⅓

$\:$

Substituting these x values in the polynomial :

★ x = 3

$\dashrightarrow~$ a(3)² + 5(3) + b = 0

$\dashrightarrow~$ 9a + 15 + b = 0

★ x = ⅓

$\dashrightarrow~$ a(⅓)² + 5(⅓) + b = 0

$\dashrightarrow~$ a(⅑) + ⁵/³ + b = 0

$\dashrightarrow~$ a(⅑)(9) + ⁵/³(9) + 9b = 0

$\dashrightarrow~$ a + 15 + 9b = 0

$\:$

Equating them both :

$\dashrightarrow~$ 9a + 15 + b = a + 15 + 9b

$\dashrightarrow~$ 9a - a + 15 - 15 + b - 9b = 0

$\dashrightarrow~$ 8a - 8b = 0

$\dashrightarrow~$ 8a = 8b

$\dashrightarrow~$ a = b

★ Hence proved!