Consider $p(x) = ax^2 - 5x + 6$ and $q(x) = bx^2 + 3x - 2$. If $p(x) \cdot q(x)$ has a $20x^3$ term and a $30x^2$ term, determine $a + b$.
The product of p(x) and q(x) is:
p(x)q(x) = (ax^2 - 5x + 6)(bx^2 + 3x - 2) = abx^4 + (3a+5b)x^3 + (2a+15+3b)x^2 + (10a+6b)x - 12
We're given that the product has a 20x^3 term and a 30x^2 term. This means that:
3a + 5b = 20 ... (1) 2a + 3b + 15 = 30 ... (2)
Solving equations (1) and (2) simultaneously, we get:
a = 5 b = 1
Therefore, a + b = 5 + 1 = 6.
