exercise:46de031dfc: Difference between revisions
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<ul style{{=}}"list-style-type:lower-alpha" | <ul style{{=}}"list-style-type:lower-alpha"> | ||
<li> | <li>Show directly that <math>\frac{2x-3}{(x-2)^2}</math> | ||
Show directly that <math>\frac{2x-3}{(x-2)^2}</math> | |||
can be written in the form | can be written in the form | ||
<math>\frac{A}{x-2} + \frac{B}{(x-2)^2}</math> | <math>\frac{A}{x-2} + \frac{B}{(x-2)^2}</math> | ||
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<math>\frac{2x-3}{(x-2)^2} = | <math>\frac{2x-3}{(x-2)^2} = | ||
\frac{2(x-2)+1}{(x-2)^2}</math>.</li> | \frac{2(x-2)+1}{(x-2)^2}</math>.</li> | ||
<li> | <li> | ||
Following the method in \ref{ex7.4.5a}, show that <math>\frac{ax+b}{(x-k)^2}</math> can always be | |||
Following the method in \ref{ex7.4.5a}, | written <math>\frac{A}{x-k} + \frac{B}{(x-k)^2}</math>, where <math>A</math> and <math>B</math> are constants.</li> | ||
show that <math>\frac{ax+b}{(x-k)^2}</math> can always be | <li>Extend the result in (b) by factoring, completing the square, and dividing to show directly that | ||
written <math>\frac{A}{x-k} + \frac{B}{(x-k)^2}</math>, | |||
where <math>A</math> and <math>B</math> are constants.</li> | |||
<li>Extend the result in | |||
completing the square, and dividing to show | |||
directly that | |||
<math display="block"> | <math display="block"> | ||
Latest revision as of 01:25, 24 November 2024
[math]
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[/math]
- Show directly that [math]\frac{2x-3}{(x-2)^2}[/math] can be written in the form [math]\frac{A}{x-2} + \frac{B}{(x-2)^2}[/math] by first writing [math]\frac{2x-3}{(x-2)^2} = \frac{2(x-2)+1}{(x-2)^2}[/math].
- Following the method in \ref{ex7.4.5a}, show that [math]\frac{ax+b}{(x-k)^2}[/math] can always be written [math]\frac{A}{x-k} + \frac{B}{(x-k)^2}[/math], where [math]A[/math] and [math]B[/math] are constants.
- Extend the result in (b) by factoring, completing the square, and dividing to show directly that
[[math]] \frac{ax^2+bx+c}{(x-k)^3} \: \mbox{can be written} \: \frac{A}{x-k}+\frac{B}{(x-k)^2}+\frac{C}{(x-k)^3} [[/math]]where [math]A[/math], [math]B[/math] and [math]C[/math] are constants.