Question
In the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not:
f(x) = 3x3 + x2 - 20x +12, g(x) = 3x - 2
f(x) = 3x3 + x2 - 20x +12, g(x) = 3x - 2
✓
Answer
Let g(x) = 0
⇒ 3x - 2 = 0
$\Rightarrow\ \text{x}=\frac{2}{3}$
$\text{f}\Big(\frac{2}{3}\Big)=3\Big(\frac{2}{3}\Big)^3+\Big(\frac{2}{3}\Big)^2-20\Big(\frac{2}{3}\Big)+12$
$=\frac{24}{27}+\frac{4}{9}-\frac{40}{3}+12$
$=\frac{24+12-360+324}{27}$
$=\frac{360-360}{27}$
$=\frac{0}{27}$
$=0$
$\because\ \text{f}\Big(\frac{2}{3}\Big)=0,$ by factor theorem, 3x - 2 is a factor of f(x).
⇒ 3x - 2 = 0
$\Rightarrow\ \text{x}=\frac{2}{3}$
$\text{f}\Big(\frac{2}{3}\Big)=3\Big(\frac{2}{3}\Big)^3+\Big(\frac{2}{3}\Big)^2-20\Big(\frac{2}{3}\Big)+12$
$=\frac{24}{27}+\frac{4}{9}-\frac{40}{3}+12$
$=\frac{24+12-360+324}{27}$
$=\frac{360-360}{27}$
$=\frac{0}{27}$
$=0$
$\because\ \text{f}\Big(\frac{2}{3}\Big)=0,$ by factor theorem, 3x - 2 is a factor of f(x).
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