# 8.4: Exercises

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

Divide by long division.

1. $$\dfrac{x^3-4x^2+2x+1}{x-2}$$
2. $$\dfrac{x^3+6x^2+7x-2}{x+3}$$
3. $$\dfrac{x^2+7x-4}{x+1}$$
4. $$\dfrac{x^3+3x^2+2x+5}{x+2}$$
5. $$\dfrac{2x^3+x^2+3x+5}{x-1}$$
6. $$\dfrac{2x^4+7x^3+x+3}{x+5}$$
7. $$\dfrac{2x^4-31x^2-13}{x-4}$$
8. $$\dfrac{x^3+27}{x+3}$$
9. $$\dfrac{3x^4+7x^3+5x^2+7x+4}{3x+1}$$
10. $$\dfrac{8x^3+18x^2+21x+18}{2x+3}$$
11. $$\dfrac{x^3+3x^2-4x-5}{x^2+2x+1}$$
12. $$\dfrac{x^5+3x^4-20}{x^2+3}$$
1. $$x^{2}-2 x-2-\dfrac{3}{x-2}$$
2. $$x^{2}+3 x-2+\dfrac{4}{x+3}$$
3. $$x+6-\dfrac{10}{x+1}$$
4. $$x^{2}+x+\dfrac{5}{x+2}$$
5. $$2 x^{2}+3 x+6+\dfrac{11}{x-1}$$
6. $$2 x^{3}-3 x^{2}+15 x-74+\dfrac{373}{x+5}$$
7. $$2 x^{3}+8 x^{2}+x+4+\dfrac{3}{x-4}$$
8. $$x^{2}-3 x+9$$
9. $$x^{3}+2 x^{2}+x+2+\dfrac{2}{3 x+1}$$
10. $$4 x^{2}+3 x+6$$
11. $$x+1-\dfrac{7 x+6}{x^{2}+2 x+1}$$
12. $$x^{3}+3 x^{2}-3 x-9+\dfrac{9 x+7}{x^{2}+3}$$

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

Find the remainder when dividing $$f(x)$$ by $$g(x)$$.

1. $$f(x)=x^3+2x^2+x-3, \quad g(x)=x-2$$
2. $$f(x)=x^3-5x+8, \quad g(x)=x-3$$
3. $$f(x)=x^5-1, \quad g(x)=x+1$$
4. $$f(x)=x^5+5x^2-7x+10, \quad g(x)=x+2$$
1. remainder $$r = 15$$
2. $$r = 20$$
3. $$r = -2$$
4. $$r = 12$$

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

Determine whether the given $$g(x)$$ is a factor of $$f(x)$$. If so, name the corresponding root of $$f(x)$$.

1. $$f(x)=x^2+5x+6, \quad g(x)=x+3$$
2. $$f(x)=x^3-x^2-3x+8, \quad g(x)=x-4$$
3. $$f(x)=x^4+7x^3+3x^2+29x+56, \quad g(x)=x+7$$
4. $$f(x)=x^{999}+1, \quad g(x)=x+1$$
1. yes, $$g(x)$$ is a factor of $$f(x)$$, the root of $$f(x)$$ is $$x = −3$$
2. $$g(x)$$ is not a factor of $$f(x)$$
3. $$g(x)$$ is a factor of $$f(x)$$, the root of $$f(x)$$ is $$x = −7$$
4. $$g(x)$$ is a factor of $$f(x)$$, the root of $$f(x)$$ is $$x = −1$$

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

Check that the given numbers for $$x$$ are roots of $$f(x)$$ (see Observation). If the numbers $$x$$ are indeed roots, then use this information to factor $$f(x)$$ as much as possible.

1. $$f(x)=x^3-2x^2-x+2, \quad x=1$$
2. $$f(x)=x^3-6x^2+11x-6, \quad x=1, x=2, x=3$$
3. $$f(x)=x^3-3x^2+x-3, \quad x=3$$
4. $$f(x)=x^3+6x^2+12x+8, \quad x=-2$$
5. $$f(x)=x^3+13x^2+50x+56, \quad x=-3, x=-4$$
6. $$f(x)=x^3+3x^2-16x-48, \quad x=2, x=-4$$
7. $$f(x)=x^5+5x^4-5x^3-25x^2+4x+20, \quad x=1, x=-1, \quad x=2, x=-2$$
1. $$f(x)=(x-2)(x-1)(x+1)$$
2. $$f(x)=(x-1)(x-2)(x-3)$$
3. $$f(x)=(x-3)(x-i)(x+i)$$
4. $$f(x)=(x+2)^{3}$$
5. $$f(x)=(x+2)(x+4)(x+7)$$
6. $$f(x)=(x-4)(x+3)(x+4)$$
7. $$f(x)=(x-2)(x-1)(x+1)(x+2)(x+5)$$

## Exercise $$\PageIndex{5}$$

Divide by using synthetic division.

1. $$\dfrac{2x^3+3x^2-5x+7}{x-2}$$
2. $$\dfrac{4x^3+3x^2-15x+18}{x+3}$$
3. $$\dfrac{x^3+4x^2-3x+1}{x+2}$$
4. $$\dfrac{x^4+x^3+1}{x-1}$$
5. $$\dfrac{x^5+32}{x+2}$$
6. $$\dfrac{x^3+5x^2-3x-10}{x+5}$$
1. $$2 x^{2}+7 x+9+\dfrac{25}{x-2}$$
2. $$4 x^{2}-9 x+12-\dfrac{18}{x+3}$$
3. $$x^{2}+2 x-7+\dfrac{15}{x+2}$$
4. $$x^{3}+2 x^{2}+2 x+2+\dfrac{3}{x-1}$$
5. $$x^{4}-2 x^{3}+4 x^{2}-8 x+16$$
6. $$x^{2}-3+\dfrac{5}{x+5}$$