Given:
The polynomial are:
[tex]ax^3-3x^2+4[/tex]
[tex]2x^3-5x+a[/tex]
If the above polynomial divided by (x-2) then they leave remainder p and q respectively.
[tex]p-2q=4[/tex]
To find:
The value of a.
Solution:
According to the remainder theorem, if a polynomial f(x) is divided by (x-c), then the remainder is f(c).
If the polynomial [tex]ax^3-3x^2+4[/tex] is divided by (x-2), then the remainder is p. So.
[tex]a(2)^3-3(2)^2+4=p[/tex]
[tex]8a-12+4=p[/tex]
[tex]8a-8=p[/tex]
If the polynomial [tex]2x^3-5x+a[/tex] is divided by (x-2), then the remainder is q. So.
[tex]2(2)^3-5(2)+a=q[/tex]
[tex]16-10+a=q[/tex]
[tex]6+a=q[/tex]
It is given that,
[tex]p-2q=4[/tex]
[tex](8a-8)-2(6+a)=4[/tex]
[tex]8a-8-12-2a=4[/tex]
[tex]6a-20=4[/tex]
Add 20 on both sides.
[tex]6a=4+20[/tex]
[tex]6a=24[/tex]
[tex]a=\dfrac{24}{6}[/tex]
[tex]a=4[/tex]
Therefore, the value of a is 4.
