Question
Find whether, or not the first polynomial is a factor of the second:
$\frac{4\text{x}^4+12\text{x}^2+15}{4\text{x}^2-5}$
$\frac{4\text{x}^4+12\text{x}^2+15}{4\text{x}^2-5}$
✓
Answer
$\frac{4\text{x}^4+12\text{x}^2+15}{4\text{x}^2-5}$
$=\frac{\text{x}^2\big(4\text{x}^2-5\big)+3\big(4\text{x}^2-5\big)+30}{4\text{x}^2-5}$
$=\Big(\text{x}^2+3\Big)+\frac{30}{4\text{x}^2-5}$
Therefore, $\left(4 x^2-5\right)$ is not a factor of $4 x^4+7 x^2+15$
$=\frac{\text{x}^2\big(4\text{x}^2-5\big)+3\big(4\text{x}^2-5\big)+30}{4\text{x}^2-5}$
$=\Big(\text{x}^2+3\Big)+\frac{30}{4\text{x}^2-5}$
Therefore, $\left(4 x^2-5\right)$ is not a factor of $4 x^4+7 x^2+15$
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