1.11B Equivalent representations of polynomial and rational expressions
For each of the following rational expressions:
Factor the numerator and denominator.
Identify the greatest common factor between the numerator and denominator.
Fully simplify the rational expression.
\dfrac{4 u}{4 u v - 20 u w}
\dfrac{6 r + 3 r s t}{3 r s - 12 r t}
Fully simplify the following rational expressions:
\dfrac{x^{2} + 7 x + 10}{x + 2}
\dfrac{x + 7}{x^{2} + 14 x + 49}
\dfrac{a^{2} + 10 a + 24}{a^{2} - 36}
\dfrac{x^{2} + 3 x - 40}{250 - 10 x^{2}}
\dfrac{5 x^{2} + 15 x - 50}{x^{2} + x - 6}
\dfrac{x^{2} - 10 x + 25}{x^{2} + 5 x - 50}
\dfrac{9 x^{2} + 9 x}{10 x^{2} + 20 x + 10}
\dfrac{2 x^{2} + 10 x - 100}{2 x + 20}
\dfrac{8 x^{2} + 2 x - 15}{\left( 4 x - 5\right)^{2}}
\dfrac{x^{3} - x^{2} - 20 x}{x^{2} + 7 x + 12}
\dfrac{8 x^{2} - 18 x + 10}{4 x^{2} + x - 5}
\dfrac{4 x^{2} + 7 x - 2}{6 + x - x^{2}}
Consider the function r(x) = \dfrac{x^2 - 5x + 6}{x - 2}. Determine the x-coordinate of the hole in the graph of the function.
Determine the equation of the slant asymptote of the function f(x) = \dfrac{2x^2 - 3x + 1}{x - 1}.
The 4 \text{th} row of Pascal’s triangle consists of the numbers 1,\, 4, \, 6, \, 4, \, 1.
Write down the numbers in the 5 \text{th} row of Pascal’s triangle.
Find the sum of the entries in the 5 \text{th} row of Pascal’s triangle.
Rewrite each element of the 5 \text{th} row of Pascal’s triangle as an {}^{n}C_{r}coefficient.
Consider the binomial \left( 4 x + 3 y\right)^{4}.
State the first term in the expansion.
State the last term in the expansion.
Simplify the rational expressions and write the solution in the form q\left(x\right) + \dfrac{r\left(x\right)}{b\left(x\right)} where r\left(x\right) is the remainder:
The volume of a box in cubic inches can be represented by the expression x^3+9x^2+20x.
If the base of the box has an area of x^2+5x square inches, find the height of the box in inches as a linear expression. Justify your answer.
Sketch an example of a possible box, labeling the three dimensions.
For each of the following, find the value(s) of k that would allow the rational expression to be simplified into a linear expression:
Given the function f(x) = \dfrac{x^2 - 4}{x - 2}, determine if the statement is true or false. Explain your reasoning.
The function has a vertical asymptote at x = 2.
The function has a horizontal asymptote at y = 0.
The function has a hole at x = 2.
The function has a horizontal asymptote at y = -2.
Mohammed claims that \dfrac{7x+16}{3x+16} will always simplify to \dfrac{7}{3}.
State the misunderstanding you think Mohammed has for him to make this claim.
Determine whether \dfrac{7x+16}{3x+16}=\dfrac{7}{3} is sometimes or never true. Justify your answer.
Qin has attempted to fully simplify a rational expression and showed his work:
| 1 | \displaystyle \dfrac{8x^2-24x}{x^2+5x-24} | \displaystyle = | \displaystyle \dfrac{8x\left(x-3\right)}{\left(x+8\right)\left(x-3\right)} | Factor the numerator and denominator |
| 2 | \displaystyle = | \displaystyle \dfrac{8x}{x+8} | Divide out \left(x+3\right) from the numerator and denominator | |
| 3 | \displaystyle = | \displaystyle \dfrac{8x}{x}+\dfrac{8x}{8} | Additive property of fractions | |
| 4 | \displaystyle = | \displaystyle 8+x | Simplify the fractions |
Identify where Qin has made an error and explain what it is.
Here is the graph of the function h(x) = \dfrac{x^2 - 4x - 5}{x - 5}. Determine the x-intercept(s) and y-intercept.
Here is a graph of a rational function. The function has a vertical asymptote at x = -6 and a slant asymptote. Determine the equation of the slant asymptote.
Use Pascal's triangle to expand the expressions:
\left(p + q\right)^{5}
\left(y + 3\right)^{4}
\left(1 + \dfrac{4}{y}\right)^{3}
\left(a - \dfrac{1}{a}\right)^{3}
Use the binomial theorem to expand the following expressions:
\left(y - 5\right)^{3}
\left(y - \dfrac{1}{3}\right)^{3}
\left(2 + y\right)^{3}
\left(u^{2} + 3 v^{2}\right)^{3}
Find the specified term for the following expansions:
Fifth term of \left(b + 2\right)^{7}.
Fifth term of \left(c + d\right)^{9}.
Tenth term of \left(u + \dfrac{1}{2}\right)^{12}.
Seventh term of \left(\dfrac{3 y}{2} - \dfrac{2}{3 y}\right)^{10}.
Eleventh term of \left( 2 x + y^{2}\right)^{13}.
Sixteenth term of \left(x - y^{5}\right)^{19}.
Middle term of \left( - 3 x^{ - 3 } + 2 y^{ - 2 }\right)^{4}.
Constant term of \left( 2 y - \dfrac{1}{y^{3}}\right)^{8}.
Consider the expansion of \left(a + b\right)^{5}.
State the coefficient of the 3 \text{rd} term in the form {}^{n}C_{r}.
Evaluate the coefficient of the 3 \text{rd} term.
By considering the symmetry of {}^{n}C_{r}, which other term has the same coefficient as the 3 \text{rd} term?
Consider the expression \dfrac{2 x + 10}{x^{2} + 5 x}.
Fully simplify the expression.
State any restrictions and determine if they will appear as a hole or an asymptote on the corresponding function.
Identify the asymptotes of the corresponding function.
Graph the corresponding function.
Consider the rational functionf\left(x\right)=\dfrac{\left(x-1\right)\left(x-2\right)}{x-3}.
Perform the long division.
Find the equation of the slant asymptote.
Graph the function f(x).
In the expansion of \left(m + k\right)^{n}, the coefficient of the 9 \text{th} term is 45.
Find the value of n.
Find the 9 \text{th} term.
Consider: \left(x + y\right)^{4} = \left(x + y\right) \left(x + y\right) \left(x + y\right) \left(x + y\right)
Determine the number of ways three factors of x and one factor of y can be chosen from the expansion to form the term which contains x^{3} y. Explain your answer.
Consider the function f(x) = \dfrac{x^3 - 3x^2 - 4x + 12}{x - 1}.
Rewrite this function in equivalent form by performing polynomial long division.
Identify the slant asymptote of this function.
How does information from the equivalent form you found help you understand the behavior of the function near the x-value of the slant asymptote?
Consider the function h(x) = \dfrac{(x + 1)^3}{x - 2}.
Expand the numerator of the function using the binomial theorem.
What does the expanded form of the function tell you about the function's behavior near x = -1?
How does this compare to the function's behavior near x = 2?
Consider the function p(x) = \dfrac{(x - 3)^2}{x^2 + 2x + 1}.
Find the slant asymptote of the function by performing polynomial long division.
Expand the numerator of the function using the binomial theorem.
How does the expanded form of the function and the slant asymptote together provide a complete picture of the function's end behavior?