Question

If a^2-2a-3 is a factor of a^4+pa^3+qa^2+12a-9 then find the value of p^2-2q-3

Answer 1

$\textbf{Factor theorem:}$

$\text{(x-a) is a factor of f(x) iff f(a) =0}$

$\textbf{Given:}$

$a^2-2a-3\;\text{is a factor of}\;a^4+p\,a^3+q\,a^2+12a-9$

$\textbf{To find:}$

$\text{The value of p^2-2q-3 }$

$\textbf{Solution:}$

$\text{Let}\;f(a)=a^4+p\,a^3+q\,a^2+12a-9$

$a^2-2a-3=(a-3)(a+1)$

$\textbf{Since (a-3) is a factor of f(a), we have f(3)=0}$

$\implies\,3^4+p\,3^3+q\,3^2+12(3)-9=0$

$\implies\,81+27\,p+9\,q+27=0$

$\implies\,27\,p+9\,q=-108$

$\implies\bf\,3\,p+q=-12$.....(1)

$\textbf{Since (a+1) is a factor of f(a), we have f(-1)=0}$

$\implies\,(-1)^4+p\,(-1)^3+q\,(-1)^2+12(-1)-9=0$

$\implies\,1-p+q-21=0$

$\implies\bf\,-p+q=20$.......(2)

$\text{Subtracting (2) from (1), we get}$

$4\,p=-32$

$\implies\bf\;p=-8$

$(2)\implies\bf\,q=12$

$\text{Now,}$

$\bf\,p^2-2q-3$

$=(-8)^2-2(12)-3$

$=64-24-3$

$=37$

$\textbf{Answer:}$

$\textbf{The value of \bf\,p^2-2q-3 is 37}$

Find more:

Find the value is of a and b, if x²-4 is a factor of ax⁴+2x³ -3x²+bx-4.

https://brainly.in/question/15904227

If x+a is a common factor of f(x)=x^(2) +px+q and g(x)=x^(2) +lx+m . Show that: a=(q-m)/(p-l)

https://brainly.in/question/3900096

If what value of a if x-3 is a factor of X3+x2-17x+a

https://brainly.in/question/5165878

Answer 2

Answer:

hey mate how are u .......

Step-by-step explanation:

ur question is answered in attachment

plzzzzz mark it as brainlest answer and thank it plzzz

thanks ❣❣❣❣❣❣❣❣