# Thread:

1. ## indeterminate forms

Could someone justify the following statement please. I'm having trouble visualizing what has been said here:

If by direct substitution of a given value c into a rational function is made and the following result occurs

$r(c)=\frac{p(c)}{q(c)}=\frac{0}{0}$

then it can be concluded that $(x-c)$ is a factor of both $p(x)$ and $q(x)$.

Anything to shed light here would be appreciated. Thanks.

2. Originally Posted by VonNemo19
Could someone justify the following statement please. I'm having trouble visualizing what has been said here:

If by direct substitution of a given value c into a rational function is made and the following result occurs

$r(c)=\frac{p(c)}{q(c)}=\frac{0}{0}$

then it can be concluded that $(x-c)$ is a factor of both $p(x)$ and $q(x)$.

Anything to shed light here would be appreciated. Thanks.
if x = c is a zero of a function, then x-c is a factor.

3. Originally Posted by VonNemo19
Could someone justify the following statement please. I'm having trouble visualizing what has been said here:

If by direct substitution of a given value c into a rational function is made and the following result occurs

$r(c)=\frac{p(c)}{q(c)}=\frac{0}{0}$

then it can be concluded that $(x-c)$ is a factor of both $p(x)$ and $q(x)$.

Anything to shed light here would be appreciated. Thanks.
I am assuming $p(x), q(x)$ are polynomials otherwise this does not make sense i.e

$\frac{\sin(\pi)}{\tan(\pi)}=\frac{0}{0}$ but neither have a factor of $x-\pi$

The above is using the fact that

If f is a polynomial and $f(c)=0$ then $(x-c)$ divides f(x)

example

$f(x)=x^2-3x+2$ note that

$f(1)=1^2-3(1)+2=0$ so by the above $(x-1)$ divides $x^2-3x+2$ i.e this factors into $f(x)=(x-1)(x-2)$

4. I can kind of see that, but if you could explain why, I'd really appreciate it.

5. Good stuff Empty dude. And yes of course they are polynomials. I kind of paraphrased what it said in the book. I'm no math writer.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 22, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9677991271018982, "perplexity_flag": "head"}