Question

Check whether x+1 is a factor of x power 4+2x power 5 +2x power 2 +x+1​

Answer 1

Answer:

Let the other factor be x  

2

+ax+b. Then

(x  

2

+2x+5)(x  

2

+ax+b)

≡x  

4

+x  

3

(2+a)+x  

2

(5+b+2a)+x(5a+2b)+5b

≡x  

4

+px  

2

+q.

Match the coefficients of like powers of x:

For x  

3

:2+a=0;∴a=−2

For x:5a+b=0;∴b=5.

For x  

2

:5+b+2a=p;∴p=6

For x  

0

:5b=q;∴q=25;

or

let y=x  

2

 so that x  

4

+px  

2

+q=y  

2

+py+q.

Let the roots of y  

2

+py+q=0 be r  

2

 and s  

2

.

Since y=x  

2

 the roots of x  

4

+px  

2

+q=0 must be ±r,±s.

Now x  

2

+2x+5 is a factor of x  

4

+px  

2

+q; consequently, one pair of roots, say r and s, must satisfy the equation x  

2

+2x+5=0. It follows that −r and −s must satisfy the equation x  

2

−2x+5=0. Therefore, the other factor must be x  

2

−2x+5.

∴(x  

2

+2x+5)(x  

2

−2x+5)≡x  

4

+6x  

2

+25≡x  

4

+px  

2

+q.

∴p=6,q=25;

or

Since the remainder must be zero (why)?

12−2p=0,p=6 and q−5p+5=0,q=25.

Step-by-step explanation: