Question

find a and b so that (z+1)and (z-1) are factors of z^4 + az^3 + 2z^2 - 3z + b .

Answer 1

Answer:

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Step-by-step explanation:

Answer 2

Given,

A polynomial: p(z) = z^4 + az^3 + 2z^2 - 3z + b

(z+1)and (z-1) are factors of p(z).

To find,

The value of a and b.

Solution,

We can simply solve this mathematical problem using the following process:

Mathematically,

If (x+a) and (x+b) are two factors of a quadratic equation p(x), then (-a) and (-b) are the two roots of the given quadratic equation and p(-a) = p(-b) = 0 {Statement-1}

Now, according to the question and statement-1;

(z+1)and (z-1) are factors of p(z)

=> (-1) and (+1) are the two roots of the given quadratic equation p(z)

=> p(+1) = p(-1) = 0

{Equation-1}

Now,

p(+1) = 0

=> (1)^4 + a(1)^3 + 2(1)^2 - 3(1) + b = 0

=> 1 + a + 2 - 3 + b = 0

=> a + b = 0

{Equation-2}

And,

p(-1) = 0

=> (-1)^4 + a(-1)^3 + 2(-1)^2 - 3(-1) + b = 0

=> 1 - a + 2 + 3 + b = 0

=> b - a = -6

{Equation-3}

Now, on adding equations 2 and 3, we get;

(a + b) + (b - a) = -6

=> 2b = -6

=> b = (-3)

Now, by substituting the value of b in equation-1, we get;

a = -b = -(-3)

=> a = (+3)

Hence, the values of a and b are (+3) and (-3), respectively.