# 27.2: Review of polynomials and rational functions

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## Exercise $$\PageIndex{1}$$

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

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

## Exercise $$\PageIndex{2}$$

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

$$21$$

## Exercise $$\PageIndex{3}$$

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

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

## Exercise $$\PageIndex{4}$$

Identify the polynomial with its graph.

1. 2. 3. 4. 1. $$f(x)=-x^{2}+2 x+1$$ graph: _______________
2. $$f(x)=-x^{3}+3 x^{2}-3 x+2$$ graph: _______________
3. $$f(x)=x^{3}-3 x^{2}+3 x+1$$ graph: _______________
4. $$f(x)=x^{4}-4 x^{3}+6 x^{2}-4 x+2$$ graph: _______________
1. $$\leftrightarrow \text { iii) }$$
2. $$\leftrightarrow \text { iv) }$$
3. $$\leftrightarrow \mathrm{i})$$
4. $$\leftrightarrow \text { ii) }$$

## 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. ## Exercise $$\PageIndex{6}$$

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

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

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

## Exercise $$\PageIndex{7}$$

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

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

## Exercise $$\PageIndex{8}$$

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

$$f(x)=(x+2)(x-5)(x-(3-2 i))(x-(3+2 i))$$ (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.

$$f(x)=\dfrac{3(x-2)(x+2)}{(x-3)(x+1)}$$ 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: ## Exercise $$\PageIndex{10}$$

Solve for $$x$$:

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