Question

Find p and q such that (x + 1 ) and (x - 1) are factors of the polynomial, $x^4+p{x}^3-3x^2+2x+q$ find the values of a and b so that the polynomial $x^3-a{x}^2-13x+b$ has $(x-1)$ and $(x+3)$ as factors.

Answer 1

$x^4+p{x}^3-3x^2+2x+2$

If (x-1) & (x+1)are the factor $\lambda$

then they will satisfy if.

x-1 = 0, x+1 = 0

x = 1 & x = -1

Let x = 1 & x = -1

$1+p-3+\alpha +q=0$

$p+2=0$ __ (1)

$1-p-3-\alpha +2=0$

$-p+q=4$ __ (2)

add both

$2q=4$

$q=2$

$p=-2$

ii) $x^3-a{x}^2-13x+b$

if (x-1) & (x+3) are factor

then they will satisfied it.

put x = 1 & x = -3

$1-a-13+b=0$

$-a+b=12$ __ (1)

$-27-9a+39+b=0$

$b=9a-12$