Question

Let f(x) be a polynomial such that f(-5)=0 then a factor of f(x) is

Answer 1

ANSWER

f(x)=f(4−x) ......replacing x by x+2

f(2+x)=f(2−x) this show f(x) is a symmetrical

The graph is like shown in figure 1

f(x) is a polynomial of degree 8 so there are 8 roots

It has two equal root which is 2 , 2 and six distinct root in form of 2+a , 2−a, 2+b, 2−b, 2+c, 2−c

Now sum of roots are 2+2+2+a+2−a+2+b+2−b+2+c+2−c

Sum of roots is 16.

I hope you are understand my solution

Answer 2

Answer :

x + 5

Note :

★ Remainder theorem : If a polynomial p(x) is divided by (x - c) , then the remainder obtained is given as R = p(c) .

★ Factor theorem :

If the remainder obtained on dividing a polynomial p(x) by (x - c) is zero , ie. if R = p(c) = 0 , then (x - c) is a factor of the polynomial p(x) .

If (x - c) is a factor of the polynomial p(x) , then the remainder obtained on dividing the polynomial p(x) by (x - c) is zero , ie. R = p(c) = 0 .

Solution :

Here ,

It is given that , f(-5) = 0 .

Since , f(-5) = 0 , thus x = -5 is obviously a zero of the polynomial f(x) .

If x = -5 , then x + 5 = 0 .

Hence ,

x + 5 is a factor of the polynomial f(x) .