Problem Statement
If $(x - 2)$ and $(x + 3)$ are factors of the polynomial $x^3 + ax^2 + bx - 30$, find the values of $a$ and $b$. Hence, factorize the polynomial completely.
Verified Solution & Marking Scheme
Apply Factor Theorem for x = 2 and x = -3
Let $f(x) = x^3 + ax^2 + bx - 30$.
1. $f(2) = 0 \implies 2^3 + a(2^2) + b(2) - 30 = 0 \implies 8 + 4a + 2b - 30 = 0 \implies 4a + 2b = 22 \implies 2a + b = 11$ ...(1)
2. $f(-3) = 0 \implies (-3)^3 + a(-3)^2 + b(-3) - 30 = 0 \implies -27 + 9a - 3b - 30 = 0 \implies 9a - 3b = 57 \implies 3a - b = 19$ ...(2)
Solve Simultaneous Equations for a and b
Add equations (1) and (2):
$(2a + b) + (3a - b) = 11 + 19 \implies 5a = 30 \implies a = 6$
Substitute $a = 6$ into (1):
$2(6) + b = 11 \implies 12 + b = 11 \implies b = -1$
Factorize Completely
The polynomial is $x^3 + 6x^2 - x - 30$.
Known factors are $(x - 2)$ and $(x + 3)$, so their product is $(x - 2)(x + 3) = x^2 + x - 6$.
Dividing $x^3 + 6x^2 - x - 30$ by $x^2 + x - 6$ yields quotient $(x + 5)$.