Question

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

Answer 1

$\textbf{Concept used:}$

$\textbf{Factor theorem:}$

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

$\text{Let f(x)=}ax^4+2x^3-3x^2+bx-4$

$\textbf{Given:}\;x^2-4\;\text{is a factor of f(x)}$

$x^2-4=x^2-2^2=(x-2)(x+2)$

$\text{Since (x-2) is a factor of f(x), f(2)=0}$

$a(2)^4+2(2)^3-3(2)^2+b(2)-4=0$

$16a+16-12+2b-4=0$

$16a+2b=0$

$8a+b=0$..........(1)

$\text{Since (x+2) is a factor of f(x), f(-2)=0}$

$a(-2)^4+2(-2)^3-3(-2)^2+b(-2)-4=0$

$16a-16-12-2b-4=0$

$16a-2b=32$

$8a-b=16$..........(2)

$\text{Adding (1) and (2), we get}$

$16a=16$

$\implies\boxed{\bf\;a=1}$

$\text{Put a=1 in (1), we get}$

$8+b=0$

$\implies\boxed{\bf\,b=-8}$

Answer 2

Answer:

a is 1

and

b is -8

I hope it helps you