Question

15. On dividing 2x3 – x2 – 9x + 13 by a polynomial g(x), the quotient and remainder were (x - 2) and (x + 5)
respectively. Find g(x).

16. Show that one and only one out of n +4, n + 7, n + 10 and n + 13 is divisible by 4.


please answer the questions​

Answer 1

Given that,

let Polynomial p(x)=3x3+4x2+5x−13

quotient g(x)=3x+10

remainder r(x)=16x−43

g(x)=?

Now, 

we know that

Euclid division lemma theorem,

p(x)=g(x)×q(x)+r(x)

3x3+4x2+5x−13=g(x)×(3x+10)+(16x−43)

3x3+4x2+5x−13−16x+43=g(x)×(3x+10)

3x3+4x2−11x+30=g(x)×(3x+10)

g(x)=3x+103x3+4x2−11x+30

now, dividing,

3x+10)3x3+4x2−11x+30(x2−2x+3−3x3+10x2_____________−6x2−11x−6x2−20x_____________9x+309x+30____________0

Hence, g(x)=x2−2x+3

This is the answer.