Answer to Question #229989 in Algebra for Benjamin

Question

If (x-2) and (x+2) are factors of the expression x^3+ax^2+bx+1 what is the sum of a and b

Answer 1

Given $x-2$ is a factor of the expression $x^3+ax^2+bx+1$

$x-2=0$

$x=2$

Substituting 2 in the equation $x^3+ax^2+bx+1$

$2^3+a(2^2)+b(2)+1=0$

$8+4a+2b+1=0$

$4a+2b=-9........................i$

Given $x+2$ is a factor of the expression $x^3+ax^2+bx+1$

$x+2=0$

$x=-2$

Substituting -2 in the equation $x^3+ax^2+bx+1$

$-2^3+a(-2^2)+b(-2)+1=0$

$-8+4a-2b+1=0$

$4a-2b=7........................ii$

Solving equation i and ii by substitution

$4a+2b=-9........................i$

$4a-2b=7.........................ii$

$4a=7+2b$

Hence $7+2b+2b=-9$

$7+4b=-9$

$4b=-16$

$b=-4$

$4a+2(-4)=-9$

$4a=-1$

$a=-\frac{1}{4}$

The sum of a and b will be

$a+b = -4-\frac{1}{4}=-\frac{17}{4}$