Question

if x-1 and X+1 are factor of ax^3+x^2-2x+b then find the value of a and b




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Answer 1

$\large\underline\purple{\bold{Solution :- }}$

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$\tt \ \: :  ⟼ Let \: f(x) \: = \: a {x}^{3} + {x}^{2} - 2x + b$

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$\large\underline{\bold{❥︎Step :- 1 }}$

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$\tt \ \: :  ⟼ Now, \: (x - 1) \: is \: a \: factor \: of \: f(x)$

$\tt\implies \:f(1) = 0$

$\tt\implies \: {a(1)}^{3 } + {(1)}^{2} - 2 \times 1 + b = 0$

$\tt\implies \:a + 1 - 2 + b = 0$

$\tt\implies \:b = 1 - a - - - (1)$

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$\large\underline{\bold{❥︎Step :- 2 }}$

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$\tt \ \: :  ⟼ Now, \: again \: (x + 1) \: is \: a \: factor \: of \: f(x)$

$\tt\implies \:f( - 1) = 0$

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

$\tt\implies \: - a + 1 + 2 + b = 0$

$\tt\implies \:b - a + 3 = 0$

$\tt\implies \:1 - a - a + 3 = 0 \: \: \: \: \: \: \: \: \: using \: (1)$

$\tt \ \: :  ⟼ 4 - 2a = 0$

$\tt\implies \:4 = 2a$

$\tt\implies \:a \: = 2$

$\tt\implies \:b \: = \: 1 - a = 1 - 2 = - 1$

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$\begin{gathered}\begin{gathered}\bf So \: - \: \begin{cases} &\sf{a \: = 2} \\ &\sf{b \: = - 1} \end{cases}\end{gathered}\end{gathered}$

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