Edna's Math Class for OpenStax
Slant asymptotes: Asymptotic Behaviour for Large #x#: #\:#Part 1
We're given the polynomial #\:p(x)\:# and the divisor #\:d(x)#: \[p(x)=25\cdot x^2+55\cdot x+72 \hspace{1.0 cm}\mbox{and}\hspace{1.0 cm}d(x) = 5\cdot x+9\hspace{0.05 cm}.\] The goal is to determine the asymptotic behaviour, as #\:\abs{x}\to\infty\:#, of the ratio \[\frac{p(x)}{d(x)}\:=\:\:\frac{25\cdot x^2+55\cdot x+72}{5\cdot x+9}\hspace{0.05 cm}.\] This problem will be accomplished in three parts. Here in Part 1, we perform the long division to determine the quotient #q(x)# and the constant remainder #r#, so that: \[\frac{p(x)}{5\cdot x+9}\:\:=\:\:q(x)+\frac{r}{5\cdot x+9}\hspace{0.05 cm}.\] In Part 2, we will use this result to determine the equation of the asymptotic behaviour, and in Part 3, we will plot the result to check our answer.
Note. One answer per blank. So, for each blank, do not type an equal sign in your answer.
| Step 1. |
Show the long division steps to divide #\:p(x)\:# by the linear term #\:d(x)=5\cdot x+9 #, #\:# to determine the quotient #q(x)# and the constant remainder #r#, so that: \[\:p(x)\:=\: \left(5\cdot x+9\right)\cdot q(x) + r\] |
| Step 2. | From Step 1, #\:#express #\:p(x)\:# in terms of the quotient #\:q(x)\:# and the remainder #\:r#, #\:# as follows: #\:p(x)\:=\: \left(5\cdot x+9\right)\cdot q(x) + r#, #\:# Put the product in the first blank and the remainder in the second blank. |
| #p(x)\:=\: 25\cdot x^2+55\cdot x+72 \:\:=# | #\large +# |
| Step 3. |
Check the answer found in Step 2. #\:# Expand the answer term-by-term and check that you do get: #\:p(x)\:=\: \left(5\cdot x+9\right)\cdot q(x) + r#. #\:# So, in the table below, calculate each product. The blue entry is the sum of the three pink entries: #\:p(x) \:=\: 5\cdot x\cdot q(x) + (9)\cdot q(x) +r#. #\:# Check that the resulting sum in the blue entry is the same as the polynomial #\:p(x)#. |
| #5\cdot x \cdot q(x)# | #\large =# | |||
| #\large +# | #(9)\cdot q(x)# | #\large =# | ||
| #5\cdot x \cdot q(x) + (9) \cdot q(x) + r# | #\large = # |
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