Question

(3x + 5) is a factor of the polynomial
(a - 1)x3 + (a + 1)x2 - (2a + 1)x - 15. Find
the value of 'a'. For this value of 'a', factorise
the given polynomial completely.​

Answer 1

Check the attachment ⭒⭑⭒⭑⭒

Answer 2

||✪✪ QUESTION ✪✪||

(3x + 5) is a factor of the polynomial (a - 1)x³ + (a + 1)x² - (2a + 1)x - 15. Find the value of 'a'. For this value of 'a', factorise the given polynomial completely. ?

|| ✰✰ ANSWER ✰✰ ||

Let f(x) = (a − 1)x³ + (a + 1)x² − (2a + 1)x − 15..

It is given that (3x + 5) is a factor of f(x). So, we can say That, remainder will be 0.

→ (3x + 5) = 0

→ 3x = (-5)

→ x = (-5/3)

Putting x = (-5/3) now, we get :-

→ f(-5/3) = 0

→ (a-1)(-5/3)³ +(a+1)(-5/3)² -(2a+1)(-5/3) -15 = 0

→ (a-1)(-125/27) + (a+1)(25/9) - (2a+1)(-5/3) -15 = 0

Taking LCM now, we get,

→ [ (a-1)(-125) + 3*25(a+1) - 9*(-5)(2a+1) -27*15 ] / 27 = 0

→ [ -125a + 125 + 75a + 75 + 90a + 45 - 405 ] = 0*27

→ ( 40a - 160 ) = 0

→ 40a = 160

→ a = (160/40)

→ a = 4.

_____________________________

Putting a = 4 in f(x) now, we get ,

→ f(x) = (4-1)x³ + (4+1)x² - (2*4 + 1)x - 15

→ f(x) = 3x³ + 5x² - 9x - 15

Now, Given That, (3x+5) is a Factor of f(x) .

So,

3x+5 ) 3x³ + 5x² - 9x - 15 ( x² - 3

subtracting 3x³ +5x²

0 -9x -15

subtracting -9x - 15

0 __

So, we can say That :-

→ 3x³ + 5x² - 9x - 15 = (3x + 5) (x² - 3)

→ 3x³ + 5x² - 9x - 15 = (3x + 5)(x² - (√3)²)

→ 3x³ + 5x² - 9x - 15 = (3x + 5)(x + √3)(x - √3) .

Hence, The Factors of Given Polynomial are (3x + 5)(x + √3)(x - √3) .

_____________________________