### About The Fundamental Theorem of Algebra:

The Fundamental Theorem of Algebra along with the Complete Factorization Theorem tells us that a polynomial of degree n will have exactly n complex solutions, although some of these solutions may be repeated.

Test Objectives
• Demonstrate the ability to write a polynomial function given certain conditions
Fundamental Theorem of Algebra Practice Test:

#1:

Instructions: Write a polynomial function of degree 3 that satisfies the given conditions.

$$a)\hspace{.2em}f(2)=-30$$ $$\text{Zeros}: 5, 4, -3$$

$$b)\hspace{.2em}f(1)=-24$$ $$\text{Zeros}: 9, -2, -1$$

#2:

Instructions: Write a polynomial function of degree 3 that satisfies the given conditions.

$$a)\hspace{.2em}f(1)=-8$$ $$\text{Zeros}: 2 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, 5$$

$$b)\hspace{.2em}f(-1)=-32$$ $$\text{Zeros}: 1\hspace{.25em}\text{multiplicity}\hspace{.25em}2, -5$$

#3:

Instructions: Write a polynomial function of degree 3 that satisfies the given conditions.

$$a)\hspace{.2em}f(2)=135$$ $$\text{Zeros}: -1 \hspace{.25em}\text{multiplicity}\hspace{.25em}3$$

$$b)\hspace{.2em}f(0)=-252$$ $$\text{Zeros}: -6, 3, -2$$

#4:

Instructions: Write a polynomial function of degree 3 that satisfies the given conditions.

$$a)\hspace{.2em}f(3)=20$$ $$\text{Zeros}: -7, -1, 2$$

$$b)\hspace{.2em}f(1)=1$$ $$\text{Zeros}: \frac{1}{2}, \frac{3}{4}, -1$$

#5:

Instructions: Write a polynomial function of degree 3 that satisfies the given conditions.

$$a)\hspace{.2em}f(-1)=-16$$ $$\text{Zeros}: 3 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, -\frac{1}{2}$$

$$b)\hspace{.2em}f(0)=-25$$ $$\text{Zeros}: -10 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, \frac{1}{4}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}f(x)=-x^3 + 6x^2 + 7x - 60$$

$$b)\hspace{.2em}f(x)=\frac{1}{2}x^3 - 3x^2 - \frac{25}{2}x - 9$$

#2:

Solutions:

$$a)\hspace{.2em}f(x)=2x^3 - 18x^2 + 48x - 40$$

$$b)\hspace{.2em}f(x)=-2x^3 - 6x^2 + 18x - 10$$

#3:

Solutions:

$$a)\hspace{.2em}f(x)=5x^3 + 15x^2 + 15x + 5$$

$$b)\hspace{.2em}f(x)=7x^3 + 35x^2 - 84x - 252$$

#4:

Solutions:

$$a)\hspace{.2em}f(x)=\frac{1}{2}x^3 + 3x^2 - \frac{9}{2}x - 7$$

$$b)\hspace{.2em}f(x)=4x^3-x^2-\frac{7}{2}x + \frac{3}{2}$$

#5:

Solutions:

$$a)\hspace{.2em}f(x)=2x^3 - 11x^2 + 12x + 9$$

$$b)\hspace{.2em}f(x)=x^3 + \frac{79}{4}x^2 + 95x - 25$$