Question

if (x+3) is a factor of x^3+ax^2-bx+6 and a+b=7 find the values of a and b

Answer 1

$p(x) = x {}^{3} + ax {}^{2} - bx + 6 \\ g(x) = x + 3 \\ by \: factor \: theorem \\ x + 3 = 0 \\ x = - 3 \\ so \: remainder = p( - 3) \\ ( - 3) {}^{ 3} + a \times ( - 3) {}^{2} - b \times - 3 + 6 = 0 \\ - 27 + 9a + 3b + 6 = 0 \\ 9a + 3b = 21 \\ 3a + b = 7 \\ b = 7 - 3a \: \: ........(i)$
It is given that a+b=7
So, from equation (i)
$a + 7 - 3a = 7 \\ - 2a = 7 - 7 \\ a = \frac{0}{ - 2} = 0$
$b = 7 - 3a = 7 - 3 \times 0 = 7 - 0 = 7$
So, a=0
and b=7.