27.2: Review of polynomials and rational functions

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Page ID: 68483

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Exercise $\PageIndex{1}$

Divide the polynomials: $\dfrac{2x^3+x^2-9x-8}{2x+3}$

Answer: $x^{2}-x-3+\dfrac{1}{2 x+3}$

Exercise $\PageIndex{2}$

Find the remainder when dividing $x^3+3x^2-5x+7$ by $x+2$.

Answer: $21$

Exercise $\PageIndex{3}$

Which of the following is a factor of $x^{400}-2x^{99}+1$: \[x-1, \quad x+1, \quad x-0 \nonumber \]

Answer: $x − 1$ is a factor, $x + 1$ is not a factor, $x − 0$ is not a factor

Exercise $\PageIndex{4}$

Identify the polynomial with its graph.

$clipboard_e0c0295236f728781dd9f837be2a680f2.png$
$clipboard_ea73579cf5c91f70adf630ab2ee0b3612.png$
$clipboard_e86c1d73b604adf1df9c67be8ac6bb649.png$
$clipboard_e1b082903b9681e6702d6ec155312e9db.png$

$f(x)=-x^{2}+2 x+1$ graph: _______________
$f(x)=-x^{3}+3 x^{2}-3 x+2$ graph: _______________
$f(x)=x^{3}-3 x^{2}+3 x+1$ graph: _______________
$f(x)=x^{4}-4 x^{3}+6 x^{2}-4 x+2$ graph: _______________

Answer

$\leftrightarrow \text { iii) }$
$\leftrightarrow \text { iv) }$
$\leftrightarrow \mathrm{i})$
$\leftrightarrow \text { ii) }$

Exercise $\PageIndex{5}$

Sketch the graph of the function: \[f(x)=x^4-10x^3-0.01x^2+0.1x \nonumber \]

What is your viewing window?
Find all roots, all maxima and all minima of the graph with the calculator.

Answer: $clipboard_e204f31b4ec80b0feba7e396e7b6b5199.png$

Exercise $\PageIndex{6}$

Find all roots of $f(x)=x^3+6x^2+5x-12$.

Use this information to factor $f(x)$ completely.

Answer: $f(x)=(x-1)(x+3)(x+4)$

Exercise $\PageIndex{7}$

Find a polynomial of degree $3$ whose roots are $0$, $1$, and $3$, and so that $f(2)=10$.

Answer: $f(x)=(-5) \cdot x(x-1)(x-3)$

Exercise $\PageIndex{8}$

Find a polynomial of degree $4$ with real coefficients, whose roots include $-2$, $5$, and $3-2i$.

Answer: $f(x)=(x+2)(x-5)(x-(3-2 i))(x-(3+2 i))$ (other correct answers are possible, depending on the choice of the first coefficient)

Exercise $\PageIndex{9}$

Let $f(x)=\dfrac{3x^2-12}{x^2-2x-3}$. Sketch the graph of $f$. Include all vertical and horizontal asymptotes, all holes, and all $x$- and $y$-intercepts.

Answer

$f(x)=\dfrac{3(x-2)(x+2)}{(x-3)(x+1)}$ has domain $D=\mathbb{R}-\{-1,3\}$, horizontal asympt. $y = 3$, vertical asympt. $x = −1$ and $x = 3$, no removable discont., $x$-intercepts at $x = −2$ and $x = 2$ and $x = 3$, $y$-intercept at $y = 4$, graph:

$clipboard_e28cf3f49f7fcd7d03fb0b4fd6823fa75.png$

Exercise $\PageIndex{10}$

Solve for $x$:

$x^4+2x< 2x^3+x^2$
$x^2+3x\geq 7$
$\dfrac{x+1}{x+4}\leq 2$

Answer

$(-1,0) \cup(1,2)$
$\left(-\infty, \dfrac{-3-\sqrt{37}}{2}\right] \cup\left[\dfrac{-3+\sqrt{37}}{2}, \infty\right)$
$(-\infty,-7] \cup(-4, \infty)$

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\)

Exercise \(\PageIndex{5}\)

Exercise \(\PageIndex{6}\)

Exercise \(\PageIndex{7}\)

Exercise \(\PageIndex{8}\)

Exercise \(\PageIndex{9}\)

Exercise \(\PageIndex{10}\)