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If two polynomials $p z = a z 3 + 4 z 2 + 3 z − 4$ and $h (z) = z 3 − 4 z + a$ are divided by the linear polynomial $g (z) = z + 1$ leaving the same remainder, then the value of $a$ is ____.

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Detailed Solution

If two polynomials

p z = a z 3 + 4 z 2 + 3 z − 4

and

h (z) = z 3 − 4 z + a

are divided by the linear polynomial

g (z) = z + 1

leaving the same remainder, then the value of

a

− 1.

The given polynomials are,

p z = a z 3 + 4 z 2 + 3 z − 4

h (z) = z 3 − 4 z + a

g (z) = z + 1

We know by the Remainder theorem that if a linear polynomial

g (z) = z − k

divides a polynomial

p (z)

of degree one or greater than one, then the remainder will be equal to

p (k) .

Equate

z + 1 = 0.

⇒ z = − 1

∴ k = − 1.

According to the question, we have,

p − 1 = h − 1

⇒ a 33 + 432 + 33 − 4 = 33 − 43 + a ⇒ 27 a + 36 + 9 − 4 = 27 − 12 + a ⇒ 27 a + 41 = 15 + a

⇒ 26 a = − 26 ⇒ a = − 1

Therefore, if two polynomials

p z = a z 3 + 4 z 2 + 3 z − 4

and

h (z) = z 3 − 4 z + a

are divided by the linear polynomial

g (z) = z + 1

leaving the same remainder, then the value of

a = − 1.

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