Question

Find 'a' and 'b' if (x²+x-2) is a factor of x³+10x²+ax+b

Answer 1

GIVEN:-

$\sf\blue{x^2+x-2}$

Splitting the middle term.

$\sf\red{x^2+2x-x-2}$

$\sf\pink{x(x+2)-1(x+2)}$

Hence the factors of p(x) are (x-1) and (x+2)

Now,

$\sf{x-1=0}$

$\sf{x=1}$

Put the value of x in given p(x).

$\sf{(1)^3+10(1)^2+a(1)+b=0}$

$\sf{1+10+a+b=0}$

$\sf{11+a+b=0}$

$\sf{a+b=-11}$.........1

Now,

x+2=0

x=-2

$\sf{(-2)^3+10(-2)^2+a(-2)+b=0}$

$\sf{-8+40-2a+b=0}$

$\sf{-2a+b=-32}$...........2

Now,

Subtracting the eq 1 and 2.

$a+b-(-2a+b)=-11-(-32)$

$a+b+2a-b=21$

$3a=21$

$a=7$

Now put the value of a in eq 1.

$a+b=-11$

$7+b=-11$

$b=-18$

Hence, The value of a and b is 7 and -18.