Notes 7 4 Partial Fractions I Composing Fractions
![Notes 7. 4 – Partial Fractions Notes 7. 4 – Partial Fractions](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-1.jpg)
![I. Composing Fractions A. ) Ex. 1– Rewrite the following expression as a single I. Composing Fractions A. ) Ex. 1– Rewrite the following expression as a single](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-2.jpg)
![II. Decomposing Fractions A. ) Steps: 1. ) If the degree of f ≥ II. Decomposing Fractions A. ) Steps: 1. ) If the degree of f ≥](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-3.jpg)
![3 a. ) If d(x) is of the form (mx + n)u 3 b. 3 a. ) If d(x) is of the form (mx + n)u 3 b.](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-4.jpg)
![B. ) For example, B. ) For example,](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-5.jpg)
![C. ) Ex. 2 - Find the partial fraction decomposition of Multiply both sides C. ) Ex. 2 - Find the partial fraction decomposition of Multiply both sides](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-6.jpg)
![Therefore Therefore](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-7.jpg)
![Solving the matrix equation gives us SUPPORT GRAPHICALLY!!! Solving the matrix equation gives us SUPPORT GRAPHICALLY!!!](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-8.jpg)
![What if we were to substitute x = 2 into the equation below? What if we were to substitute x = 2 into the equation below?](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-9.jpg)
![Now, substitute x = -3 into the equation. Now, substitute x = -3 into the equation.](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-10.jpg)
![D. ) Ex. 3 - Find the partial fraction decomposition of D. ) Ex. 3 - Find the partial fraction decomposition of](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-11.jpg)
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-12.jpg)
![Try this, substitute x = 0 into the equation below. Try this, substitute x = 0 into the equation below.](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-13.jpg)
![Now, substitute x = -3 into the equation below. We can now choose any Now, substitute x = -3 into the equation below. We can now choose any](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-14.jpg)
![III. Denominators of Irreducible Quadratic Factors A. ) If d(x) is of the form III. Denominators of Irreducible Quadratic Factors A. ) If d(x) is of the form](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-15.jpg)
![B. ) Ex. 4 – Write the partial fraction decomposition of B. ) Ex. 4 – Write the partial fraction decomposition of](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-16.jpg)
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-17.jpg)
![Unfortunately, there is no shortcut method for denominators with irreducible quadratics. C. ) Ex. Unfortunately, there is no shortcut method for denominators with irreducible quadratics. C. ) Ex.](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-18.jpg)
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-19.jpg)
![A rref(A) A rref(A)](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-20.jpg)
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-21.jpg)
![IV. Applications of Partial Fractions A. ) Ex. 6 - Graph the following function IV. Applications of Partial Fractions A. ) Ex. 6 - Graph the following function](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-22.jpg)
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-23.jpg)
![Now, lets look at the partial fraction decomposition of the function and the graphs Now, lets look at the partial fraction decomposition of the function and the graphs](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-24.jpg)
![First Piece - Models the end behavior of the graph First Piece - Models the end behavior of the graph](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-25.jpg)
![2 nd Piece - Models the behavior of the graph near the vertical asymptote 2 nd Piece - Models the behavior of the graph near the vertical asymptote](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-26.jpg)
![3 rd Piece - Models the behavior of the graph near the vertical asymptote 3 rd Piece - Models the behavior of the graph near the vertical asymptote](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-27.jpg)
![Combining the 3 pieces gives us this… Or, more precisely, something like this… Combining the 3 pieces gives us this… Or, more precisely, something like this…](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-28.jpg)
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-29.jpg)
![B. ) Ex. 7 – Explain in sentences the three different parts to the B. ) Ex. 7 – Explain in sentences the three different parts to the](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-30.jpg)
![First, the graph’s end-behavior asymptote will be modeled by the graph y = -x First, the graph’s end-behavior asymptote will be modeled by the graph y = -x](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-31.jpg)
![Combining the three separate parts gives us a graph that looks like Combining the three separate parts gives us a graph that looks like](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-32.jpg)
- Slides: 32
![Notes 7 4 Partial Fractions Notes 7. 4 – Partial Fractions](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-1.jpg)
Notes 7. 4 – Partial Fractions
![I Composing Fractions A Ex 1 Rewrite the following expression as a single I. Composing Fractions A. ) Ex. 1– Rewrite the following expression as a single](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-2.jpg)
I. Composing Fractions A. ) Ex. 1– Rewrite the following expression as a single rational expression.
![II Decomposing Fractions A Steps 1 If the degree of f II. Decomposing Fractions A. ) Steps: 1. ) If the degree of f ≥](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-3.jpg)
II. Decomposing Fractions A. ) Steps: 1. ) If the degree of f ≥ the degree of d, then 2. ) Factor d(x) into a product of factors of the form (mx + n)u or (ax 2 + bx + c)v when (ax 2 + bx + c) is irreducible.
![3 a If dx is of the form mx nu 3 b 3 a. ) If d(x) is of the form (mx + n)u 3 b.](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-4.jpg)
3 a. ) If d(x) is of the form (mx + n)u 3 b. ) If d(x) is of the form (ax 2 + bx + c)v
![B For example B. ) For example,](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-5.jpg)
B. ) For example,
![C Ex 2 Find the partial fraction decomposition of Multiply both sides C. ) Ex. 2 - Find the partial fraction decomposition of Multiply both sides](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-6.jpg)
C. ) Ex. 2 - Find the partial fraction decomposition of Multiply both sides by the LCD
![Therefore Therefore](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-7.jpg)
Therefore
![Solving the matrix equation gives us SUPPORT GRAPHICALLY Solving the matrix equation gives us SUPPORT GRAPHICALLY!!!](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-8.jpg)
Solving the matrix equation gives us SUPPORT GRAPHICALLY!!!
![What if we were to substitute x 2 into the equation below What if we were to substitute x = 2 into the equation below?](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-9.jpg)
What if we were to substitute x = 2 into the equation below?
![Now substitute x 3 into the equation Now, substitute x = -3 into the equation.](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-10.jpg)
Now, substitute x = -3 into the equation.
![D Ex 3 Find the partial fraction decomposition of D. ) Ex. 3 - Find the partial fraction decomposition of](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-11.jpg)
D. ) Ex. 3 - Find the partial fraction decomposition of
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-12.jpg)
![Try this substitute x 0 into the equation below Try this, substitute x = 0 into the equation below.](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-13.jpg)
Try this, substitute x = 0 into the equation below.
![Now substitute x 3 into the equation below We can now choose any Now, substitute x = -3 into the equation below. We can now choose any](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-14.jpg)
Now, substitute x = -3 into the equation below. We can now choose any value for x and substitute it into one of the equations along with A 1 and A 3 and solve for A 2.
![III Denominators of Irreducible Quadratic Factors A If dx is of the form III. Denominators of Irreducible Quadratic Factors A. ) If d(x) is of the form](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-15.jpg)
III. Denominators of Irreducible Quadratic Factors A. ) If d(x) is of the form (ax 2 + bx + c)v Since subscripts can get confusing, we can simply use different letters.
![B Ex 4 Write the partial fraction decomposition of B. ) Ex. 4 – Write the partial fraction decomposition of](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-16.jpg)
B. ) Ex. 4 – Write the partial fraction decomposition of
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-17.jpg)
![Unfortunately there is no shortcut method for denominators with irreducible quadratics C Ex Unfortunately, there is no shortcut method for denominators with irreducible quadratics. C. ) Ex.](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-18.jpg)
Unfortunately, there is no shortcut method for denominators with irreducible quadratics. C. ) Ex. 5 – Find the partial fraction decomposition of
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-19.jpg)
![A rrefA A rref(A)](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-20.jpg)
A rref(A)
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-21.jpg)
![IV Applications of Partial Fractions A Ex 6 Graph the following function IV. Applications of Partial Fractions A. ) Ex. 6 - Graph the following function](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-22.jpg)
IV. Applications of Partial Fractions A. ) Ex. 6 - Graph the following function on your calculator. Before we graph it, what can you tell me about what the graph will look like?
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-23.jpg)
![Now lets look at the partial fraction decomposition of the function and the graphs Now, lets look at the partial fraction decomposition of the function and the graphs](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-24.jpg)
Now, lets look at the partial fraction decomposition of the function and the graphs of the individual pieces.
![First Piece Models the end behavior of the graph First Piece - Models the end behavior of the graph](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-25.jpg)
First Piece - Models the end behavior of the graph
![2 nd Piece Models the behavior of the graph near the vertical asymptote 2 nd Piece - Models the behavior of the graph near the vertical asymptote](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-26.jpg)
2 nd Piece - Models the behavior of the graph near the vertical asymptote x =-2
![3 rd Piece Models the behavior of the graph near the vertical asymptote 3 rd Piece - Models the behavior of the graph near the vertical asymptote](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-27.jpg)
3 rd Piece - Models the behavior of the graph near the vertical asymptote x =2
![Combining the 3 pieces gives us this Or more precisely something like this Combining the 3 pieces gives us this… Or, more precisely, something like this…](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-28.jpg)
Combining the 3 pieces gives us this… Or, more precisely, something like this…
![](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-29.jpg)
![B Ex 7 Explain in sentences the three different parts to the B. ) Ex. 7 – Explain in sentences the three different parts to the](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-30.jpg)
B. ) Ex. 7 – Explain in sentences the three different parts to the graph of the following function. Then, graph the function.
![First the graphs endbehavior asymptote will be modeled by the graph y x First, the graph’s end-behavior asymptote will be modeled by the graph y = -x](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-31.jpg)
First, the graph’s end-behavior asymptote will be modeled by the graph y = -x +1. Second, the graph of the function at x = -3 will closely resemble the behavior of the graph of the hyperbola with its branches in the upper right and lower left quadrants. Third, the graph of the function at x = 2 will closely resemble the behavior of the graph of the hyperbola with its branches in the upper left and lower right quadrants.
![Combining the three separate parts gives us a graph that looks like Combining the three separate parts gives us a graph that looks like](https://slidetodoc.com/presentation_image_h/1f91361b29ef656beb1e643fe5030eea/image-32.jpg)
Combining the three separate parts gives us a graph that looks like
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