Question

decide if (x-3) is a factor of 3x³+10x²-x-12

Explanation:

Step1: Recall the Factor Theorem

The Factor Theorem states that if \((x - a)\) is a factor of a polynomial \(f(x)\), then \(f(a)=0\). Here, \(a = 3\) (since the factor is \((x - 3)\)) and the polynomial is \(f(x)=3x^{3}+10x^{2}-x - 12\). So we need to evaluate \(f(3)\).

Step2: Substitute \(x = 3\) into the polynomial

\[

$$\begin{align*} f(3)&=3\times(3)^{3}+10\times(3)^{2}-3 - 12\\ &=3\times27+10\times9-3 - 12\\ &=81 + 90-3 - 12\\ &=(81 + 90)-(3 + 12)\\ &=171-15\\ &=156 \end{align*}$$

\]

Since \(f(3)=156
eq0\), by the Factor Theorem, \((x - 3)\) is not a factor of \(3x^{3}+10x^{2}-x - 12\).

Answer:

No, it is not a factor!