The factor theorem of algebra states that if a polynomial Px

The factor theorem of algebra states that if a polynomial, P(x), is divided by xa, then the remainder is P(a). Verify the remainder theorem by showing that when x27x+16 is divided by x4 the remainder is the same as P(4).

Remainder for x27x+16x4:

P(4)= The factor theorem of algebra states that if a polynomial, P(x), is divided by xa, then the remainder is P(a). Verify the remainder theorem by showing that when x27x+16 divided by x4 the remainder is the same as P(4).

Remainder for x27x+16x4:  

P(4)=

Solution

f(x) =x²-7x+16
When it is divided by (x-a) remainder is f(a)

Hence when f(x) is divided by x-4 the remainder will be f(4) = 4² -7*4 +16 = 16-28+16 = 4;
Now actually dividing x² -7x + 16 by (x-4)
we can write the numerator as (x-4)(x-p) +q = x² -x(4+p) +4p+q
comparing the coeffecients we get 4+p =7 so p=7-4 =3;
4p+q=16 so 4*3+q=16 so q=16-12=4
Thus p=3 & q=4
Thus x² -7x + 16 = (x-4)(x-3) +4
Now dividing this by (x-4) we get =

[(x-4)(x-3) +4 ] /(x-4) = (x-3)(x-4) / (x-4) + 4/(x-4) = (x-3) + 4/(x-4) Hence the remainder is 4

As has been proven through the factor theorem since we get f(4) =4