Problem Statement
Using the Factor Theorem, show that $(x - 2)$ is a factor of $P(x) = 2x^3 + x^2 - 13x + 6$. Hence, factorize $P(x)$ completely.
Verified Solution & Marking Scheme
Verify (x - 2) is a factor by evaluating P(2)
Substitute $x = 2$ into $P(x)$:
$$P(2) = 2(2)^3 + (2)^2 - 13(2) + 6$$
$$= 2(8) + 4 - 26 + 6 = 16 + 4 - 26 + 6 = 26 - 26 = 0$$
Since $P(2) = 0$, by Factor Theorem, $(x - 2)$ is a factor of $P(x)$.
Divide P(x) by (x - 2) using polynomial division
Dividing $2x^3 + x^2 - 13x + 6$ by $(x - 2)$:
$$2x^3 + x^2 - 13x + 6 = (x - 2)(2x^2 + 5x - 3)$$
Factorize the quadratic quotient 2x² + 5x - 3
$$2x^2 + 5x - 3 = 2x^2 + 6x - x - 3$$
$$= 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3)$$
Write complete linear factorization
$$P(x) = (x - 2)(2x - 1)(x + 3)$$