Question

find the value of a and b such that x^2-2x-3 is a factor of x^3-3x^2+ax-b​

Answer 1

Answer:

b = 6

a = -2

Step-by-step explanation:

here,

x² - 2x - 3

x² + x - 3x - 3

x(x + 1) -3(x + 1)

(x - 3)(x + 1)

x = 3,-1

since it is factor of

x³ - 3x² + ax - b

if we put value of x in it equals to zero

if x = 3

3³ - 3(3)² + 3a - b = 0

27 - 27 + 3a - b = 0

3a - b = 0 ...(1)

now,

if x = -1

(-1)³ - 3(-1)² - a - b = 0

-1 - 3 - a - b = 0

a - b = 4 ...(2)

(1) - (2)

3a - b - (a - b) = 0 - 4

3a - b - a + b = -4

2a = -4

a = -2

puttingthe value of a on (2)

a - b = -4

2 - b = -4

-b = -4 - 2

b = 6

a = -2

Answer 2

$\huge{\bigstar{\bold{\red{\underline{\underline{Solution:}}}}}}$

$\sf{\large{{x}^{2} - 2x - 3}}$

$\sf{\large{ {x}^{2} + x - 3x - 3}}$

$\sf{\large{x(x + 1) - 3(x + 1)}}$

$\sf{\large{(x - 3)(x + 1)}}$

$\sf{\large{x = 3, - 1}}$

$\text{\large{Since, \:it \:is \:a\: factor\: of}}$

$\sf{\large{{x}^{3} - {3x}^{2} + ax - b}}$

$\text{\large{We\:put\: value\:of\:x\:in\:it\:equals\:to\:zero}}$

$\bf{\large{\pink{if\:x=3}}}$

$\bf{\large{\pink{ {3}^{3} - 3( {3)}^{2} + 3a - b = 0}}}$

$\bf{\large{\pink{27 - 27 + 3a - b = 0}}}$

$\tt{\large{3a - b = 0...x(1)}}$

$\tt{\large{Now,}}$

$\tt{\large{if\:x=-1}}$

$\tt{\large{ {(-1)}^{3} -3{(-1)}^{2}-a-b=0}}$

$\tt{\large{-1-3-a-b=0}}$

$\sf{\large{a - b = - 4....(2)}}$[/tex]

$\sf{\large{(1) - (2)}}$[/tex]

$\sf{\large{3a - b - (a + b) = 0 - 4}}$

$\sf{\large{3a - b - a + b = - 4}}$[/tex]

$\sf{\large{2 a = - 4}}$

$\sf{\large{a = - 2}}$

$\text{\large{Putting\:the\: value\: of\:a\:on}}$

$\bf{\large{\blue{a - b = - 4}}}$

$\bf{\large{\blue{2 - b = - 4}}}$

$\bf{\large{\blue{- b = - 4 - 2}}}$

$\bf{\large{\blue{b = 6}}}$

$\bf{\large{\blue{ a = - 2}}}$