A-Level Edexcel Maths Pure Differentiation: Given that $(x + 3)$ is a factor

Question

Given that $(x + 3)$ is a factor of $f(x)$, find the value of the constant $a$.

Let $ f(x) = 3x^3 + 2ax^2 - 4x + 5a $.

Since $(x + 3)$ is a factor, applying the Factor Theorem, we have:
$$ f(-3) = 0 $$

Substituting $-3$ into the function:
$$ f(-3) = 3(-3)^3 + 2a(-3)^2 - 4(-3) + 5a = 0 $$

Calculating the values:
$$ = 3(-27) + 2a(9) + 12 + 5a $$
$$ = -81 + 18a + 12 + 5a = 0 $$
$$ = 23a - 69 = 0 $$
$$ 23a = 69 $$
$$ a = 3 $$

Given that $(x + 3)$ is a factor of $f(x)$, find the value of the constant $a$ - Edexcel - A-Level Maths Pure - Question 3 - 2019 - Paper 1