# 10.3: Exercises

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

1. Find all rational roots of $$f(x)=2x^3-3x^2-3x+2$$.
2. Find all rational roots of $$f(x)=3x^3-x^2+15x-5$$.
3. Find all rational roots of $$f(x)=6x^3+7x^2-11x-12$$.
4. Find all real roots of $$f(x)=6x^4+25x^3+8x^2-7x-2$$.
5. Find all real roots of $$f(x)=4x^3+9x^2+26x+6$$.
1. $$x=-1, x=2, x=\dfrac{1}{2}$$
2. $$x=\dfrac{1}{3}$$
3. $$x=\dfrac{-3}{2}, x=-1, x=\dfrac{4}{3},$$
4. $$x=\dfrac{1}{2}, x=\dfrac{-2}{3}, x=-2+\sqrt{3}, x=-2-\sqrt{3}$$
5. $$x=-\dfrac{1}{4}$$

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

Find a root of the polynomial by guessing possible candidates of the root.

1. $$f(x)=x^5-1$$
2. $$f(x)=x^4-1$$
3. $$f(x)=x^3-27$$
4. $$f(x)=x^3+1000$$
5. $$f(x)=x^4-81$$
6. $$f(x)=x^3-125$$
7. $$f(x)=x^{5}+32$$
8. $$f(x)=x^{777}-1$$
9. $$f(x)=x^2+64$$
1. $$x = 1$$
2. $$x = 1$$ or $$x = −1$$
3. $$x = 3$$
4. $$x = −10$$
5. $$x = 3$$ or $$x = −3$$
6. $$x = 5$$
7. $$x = −2$$
8. $$x = 1$$
9. $$x = 8i$$ or $$x = −8i$$

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

Find the roots of the polynomial and use it to factor the polynomial completely.

1. $$f(x)=x^3-7x+6$$
2. $$f(x)=x^3-x^2-16x-20$$
3. $$f(x)=x^4-5x^2+4$$
4. $$f(x)=x^3+x^2-5x-2$$
5. $$f(x)=2x^3+x^2-7x-6$$
6. $$f(x)=12x^3+49x^2-2x-24$$
7. $$f(x)=x^4-1$$
8. $$f(x)=x^5-6x^4+8x^3+6x^2-9x$$
9. $$f(x)=x^3-27$$
10. $$f(x)=x^4+2x^2-15$$
1. $$f(x)=(x-2)(x-1)(x+3)$$
2. $$f(x)=(x-5)(x+2)^{2}$$
3. $$f(x)=(x-1)(x+1)(x-2)(x+2)$$
4. $$f(x)=(x-2)\left(x-\dfrac{-3+\sqrt{5}}{2}\right)\left(x-\dfrac{-3-\sqrt{5}}{2}\right)$$
5. $$f(x)=2\left(x+\dfrac{3}{2}\right)(x+1)(x-2)$$
6. $$f(x)=12\left(x-\dfrac{2}{3}\right)\left(x+\dfrac{3}{4}\right)(x+4)$$
7. $$f(x)=(x-1)(x+1)(x-i)(x+i)$$
8. $$f(x)=x(x-1)(x+1)(x-3)^{2}$$
9. $$f(x)=(x-3)\left(x-\dfrac{-3+3 \sqrt{3} \cdot i}{2}\right)\left(x-\dfrac{-3-3 \sqrt{3} \cdot i}{2}\right)$$
10. $$f(x)=(x-\sqrt{3})(x+\sqrt{3})(x-\sqrt{5} \cdot i)(x+\sqrt{5} \cdot i)$$

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

Find the exact roots of the polynomial; write the roots in simplest radical form, if necessary. Sketch a graph of the polynomial with all roots clearly marked.

1. $$f(x)=x^3-2x^2-5x+6$$
2. $$f(x)=x^3+5x^2+3x-4$$
3. $$f(x)=-x^3+5x^2+7x-35$$
4. $$f(x)=x^3+7x^2+13x+7$$
5. $$f(x)=2x^3-8x^2-18x-36$$
6. $$f(x)=x^4-4x^2+3$$
7. $$f(x)=-x^4+x^3+24x^2-4x-80$$
8. $$f(x)=7x^3-11x^2-10x+8$$
9. $$f(x)=-15x^3+41x^2+15x-9$$
10. $$f(x)=x^4-6x^3+6x^2+4x$$

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

Find a polynomial $$f$$ that fits the given data.

1. $$f$$ has degree $$3$$. The roots of $$f$$ are precisely $$2$$, $$3$$, $$4$$. The leading coefficient of $$f$$ is $$2$$.
2. $$f$$ has degree $$4$$. The roots of $$f$$ are precisely $$-1$$, $$2$$, $$0$$, $$-3$$. The leading coefficient of $$f$$ is $$-1$$.
3. $$f$$ has degree $$3$$. $$f$$ has roots $$-2$$, $$-1$$, $$2$$, and $$f(0)=10$$.
4. $$f$$ has degree $$4$$. $$f$$ has roots $$0$$, $$2$$, $$-1$$, $$-4$$, and $$f(1)=20$$.
5. $$f$$ has degree $$3$$. The coefficients of $$f$$ are all real. The roots of $$f$$ are precisely $$2+5i$$, $$2-5i$$, $$7$$. The leading coefficient of $$f$$ is $$3$$.
6. $$f$$ has degree $$3$$. The coefficients of $$f$$ are all real. $$f$$ has roots $$i$$, $$3$$, and $$f(0)=6$$.
7. $$f$$ has degree $$4$$. The coefficients of $$f$$ are all real. $$f$$ has roots $$5+i$$ and $$5-i$$ of multiplicity $$1$$, the root $$3$$ of multiplicity $$2$$, and $$f(5)=7$$.
8. $$f$$ has degree $$4$$. The coefficients of $$f$$ are all real. $$f$$ has roots $$i$$ and $$3+2i$$.
9. $$f$$ has degree $$6$$. $$f$$ has complex coefficients. $$f$$ has roots $$1+i$$, $$2+i$$, $$4-3i$$ of multiplicity $$1$$ and the root $$-2$$ of multiplicity $$3$$.
10. $$f$$ has degree $$5$$. $$f$$ has complex coefficients. $$f$$ has roots $$i$$, $$3$$, $$-7$$ (and possibly other roots).
11. $$f$$ has degree $$3$$. The roots of $$f$$ are determined by its graph:

1. $$f$$ has degree $$4$$. The coefficients of $$f$$ are all real. The leading coefficient of $$f$$ is 1. The roots of $$f$$ are determined by its graph: (see Section 9.3).

1. $$f$$ has degree $$4$$. The coefficients of $$f$$ are all real. $$f$$ has the following graph:

1. $$f(x)=2(x-2)(x-3)(x-4)$$
2. $$f(x)=(-1) \cdot x(x-2)(x+1)(x+3)$$
3. $$f(x)=\left(-\dfrac{5}{2}\right) \cdot(x-2)(x+2)(x+1)$$
4. $$f(x)=-2 \cdot x(x-2)(x+1)(x+4)$$
5. $$f(x)=3(x-7)(x-(2+5 i))(x-(2-5 i))$$
6. $$f(x)=(-2) \cdot(x-i)(x+i)(x-3)$$
7. $$f(x)=\frac{7}{4} \cdot(x-(5+i))(x-(5-i))(x-3)^{2}$$
8. $$f(x)=(x-i)(x+i)(x-(3+2 i))(x-(3-2 i))$$ (other correct answers are possible, depending on the choice of the first coefficient)
9. $$f(x)=(x-(1+i))(x-(2+i))(x-(4-3 i))(x+2)^{3}$$ (other correct answers are possible, depending on the choice of the first coefficient)
10. $$f(x)=(x-i)(x-3)(x+7)^{2}$$ (other correct answers are possible, depending on the choice of the first coefficient and the fourth root)
11. $$f(x)=(x-2)(x-3)(x-4)$$ (other correct answers are possible, depending on the choice of the first coefficient)
12. $$f(x)=(x-1)^{2}(x-3)^{2}$$
13. $$f(x)=-x(x-1)(x-3)(x-4)$$ (other correct answers are possible, depending on the choice of the first coefficient)

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