Question

Find the value of a and b so that x+1 and x-1 are factors of x + a.x² + 2x²-3x+b


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Answer 1

Correct question :

Find the value of a and b so that, (x + 1) and (x – 1) are the factors of x⁴ + ax³ – 3x² + 2x + b.

Answer:

The values of a and b are – 2 and 2 and so that (x + 1) and (x – 1) are the factors of x⁴ + ax³ – 3x² + 2x + b.

Step-by-step explanation:

Given that, (x + 1) and (x – 1) are the factors of the polynomial x⁴ + ax³ – 3x² + 2x + b.

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Now, this means that for p(x) = x⁴ + ax³ – 3x² + 2x + b, p(–1) and p(1) will always be 0.

This can be said from the factor theorem.

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★ C A S E 1 : p ( – 1 ) :

Substituting p(–1) in the polynomial :

$\dashrightarrow\sf~~ x^4 + ax^3 - 3x^2 + 2x + b=0$

$\sf\dashrightarrow~~ (-1)^4 + a(-1)^3 - 3(-1)^2 +2(-1) + b=0$

$\dashrightarrow\sf~~ 1 + a(-1) -3-2 + b=0$

$\dashrightarrow\sf~~ 1 - a - 5 + b =0$

$\dashrightarrow\sf~~-a+b-4=0$

$\dashrightarrow\pmb{\sf~~{ -a+b=4}}$

★ C A S E 2 : p ( 1 ) :

Substituting p(1) in the polynomial :

$\dashrightarrow\sf~~ x^4 + ax^3 -3x^2 +2x + b=0$

$\dashrightarrow\sf~~ (1)^4+a(1)^3-3(1)^2+2(1)+b=0$

$\dashrightarrow\sf~~ 1+a-3+2+b=0$

$\dashrightarrow\pmb{\sf~~{ a+b=0}}$

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Now, the two linear equations are :

$\dashrightarrow\sf~~ -a+b=4\dots\dots$⑴

$\dashrightarrow\sf~~ a+b=0\dots\dots$⑵

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Getting the value of b from ⑴ :

$\dashrightarrow\sf~~ -a+b=4$

$\dashrightarrow\sf~~ b = 4+a$

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Substituting this value of b in ⑵ :

$\dashrightarrow\sf~~a+b=0$

$\dashrightarrow\sf~ ~a+(4+a)=0$

$\dashrightarrow\sf~~2a+4=0$

$\dashrightarrow\sf~~ 2a =-4$

$\dashrightarrow\pmb{\sf~~ {a = -2}}$

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Substituting this value of a in ⑴ to get b :

$\dashrightarrow\sf~~ a+b=0$

$\dashrightarrow\sf~~-2+b=0$

$\dashrightarrow\pmb{\sf~~{ b=2}}$

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Therefore, the values of a and b so that (x + 1) and (x – 1) are the factors of x⁴ + ax³ – 3x² + 2x + b = 0 are – 2 and 2 respectively.