Question

Find the values of a and b so that x^4+x³+8x+ax-b is divisible by x²+1

Answer 1

AnsweR :

GiveN :

$\tt{p(x) = {x}^{4} + {x}^{3} + 8x + ax - b }$

P(x) is divisible by x² + 1

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Required to Find :

Values of ' a ' and ' b '

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ExplanatioN :

In the question it is given that ;

p(x) is divisible by x²+1 .

So,

x² + 1 is the factor of p(x) .

The word factor means it is the multiple of the given polynomial expression .

Similarly, when the value factor is substituted in the p(x) then the remainder becomes zero .

So, using this above idea we can solve this question .

Here, equal the factor value with zero .

The number which we must substitute is an imaginary number .

which is called as iota

so, using that we have to solve this question

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Solution :

Given :

$\tt{p(x) = {x}^{4} + {x}^{3} + 8x + ax - b }$

It is given that it is divisble by x²+1 .

Now,

Equal the value of this with zero .

So,

$\longrightarrow{{x}^{2} + 1 = 0}$

$\longrightarrow{{x}^{2} = - 1}$

$\longrightarrow{x = \sqrt{-1}}$

$\rightarrow{ x = i}$

Hence, the values of x is i

Here i is called as iota which is an imaginary number which belong to the complex number system .

So, substitute this in p(x)

p(i) = 0

i⁴ + i³ + 8i + ai - b = 0

1 - i + 8i + ai - b = 0

1 + 7i + ai - b = 0

(1 - b)+i(a + 7)= 0

Here compare the real and imaginary part on both sides .

So,

1 - b = 0

=> b = 1

Similarly,

a + 7 = 0

=> a = -7

$\Rightarrow{\boxed{\therefore{\tt{ Value \;of \;" a "\; and\; " b "\; = -7 \;and \;1}}}}$

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✅ Hence Solved .

Answer 2

Answer:

a = -7

b = 1

Step-by-step explanation:

Given that

p(x) = x⁴ + x³ + 8x + ax - b is

divisible by x² + 1.

Here we can apply Remainder Theorem which states that if p(x) is divisible by x²+ 1 , then it should satisfy

x² + 1 = 0

⇒x²  = -1

⇒ x = √(-1)

⇒ x = i

Where 'i' is an imaginary number called IOTA.

Now  putting x = i in given polynomial p(x)

We have,

p(i) = 0

i⁴ + i³ + 8i + ai - b = 0

1 - i +8i + ai - b = 0

1 + 7i + ai - b = 0

(1 - b) + i(a + 7) = 0

Comparing the real and imaginary parts on both sides, We get,

1 - b = 0

⇒ b = 1

and,

a + 7 = 0

⇒ a = -7

Note:-  1) i² =  -1

           

           2) i³ = -i

           3) i⁴ = 1