Math, asked by Chetana35, 9 months ago

one factor of x^4+x^2+1​

Answers

Answered by amanpatel8084
1

Step-by-step explanation:

x

4

+

x

2

+

1

=

(

x

2

+

x

+

1

)

(

x

2

x

+

1

)

To find this, first notice that

x

4

+

x

2

+

1

>

0

for all (real) values of

x

. So there are no linear factors, only quadratic ones.

x

4

+

x

2

+

1

=

(

a

x

2

+

b

x

+

c

)

(

d

x

2

+

e

x

+

f

)

Without bothering to multiply this out fully just yet, notice that the coefficient of

x

4

gives us

a

d

=

1

. We might as well let

a

=

1

and

d

=

1

.

...

=

(

x

2

+

b

x

+

c

)

(

x

2

+

e

x

+

f

)

Next, the coefficient of

x

3

gives us

b

+

e

=

0

, so

e

=

b

.

...

=

(

x

2

+

b

x

+

c

)

(

x

2

b

x

+

f

)

The constant term gives us

c

f

=

1

, so either

c

=

f

=

1

or

c

=

f

=

1

. Let's try

c

=

f

=

1

.

...

=

(

x

2

+

b

x

+

1

)

(

x

2

b

x

+

1

)

Note that the coefficient of

x

will vanish nicely when these are multiplied out.

Finally notice that the coefficient of

x

2

is

(

1

b

2

+

1

)

=

2

b

2

, giving us

1

=

2

b

2

, thus

b

2

=

1

, so

b

=

1

or

b

=

1

.

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