# f(x) = -x^4 + 4x^3 + ax^2 + bx + 9 This function has a factor of x-1 This function when divided by x+4 has a remainder that is five times larger than the remainder when divided by x+5. Find out the...

f(x) = -x^4 + 4x^3 + ax^2 + bx + 9

This function has a factor of x-1

This function when divided by x+4 has a remainder that is five times larger than the remainder when divided by x+5.

Find out the values of a and b.

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Sorry, this function when divided by x+4 has a remainder that is five times larger than the remainder when divided by x+2

f(x) = -x^4 + 4x^3 + ax^2 + bx + 9

This function has a factor of x-1

This function when divided by x+4 has a remainder that is five times larger than the remainder when divided by x+2.

Find out the values of a and b.

The f(x) has factor x-1. Therefore f(1)=0

f(1)=-1+4+a+b+9=0

a+b=-12 (i)

The remainder when f(x) divided by x+4 is f(-4).

f(-4)=-256-256+16a-4b+9

=16a-4b-503

The remainder when f(x) divided by x+2 is f(-2).

f(-2)=-16-16+4a-2b+9

=4a-2b-23

But f(-4)=5 f(-2)

16a-4b-503= 5(4a-2b-23)

16a-4b-20a+10b=-115+503

-4a+6b=388 (ii)

To solve (i) and (ii), multiply (i) by 4 and add to (ii), we have

b=34

Substitute b=34 in (i), we have

a=-46

Thus a=-46 and b=34.