### About The Conjugate Zeros Theorem:

The conjugate zeros theorem tells us that if our polynomial function has real number coefficients, then our complex zeros will always come in pairs. So if a + bi is a zero, then a - bi is also a zero.

Test Objectives
• Demonstrate the ability to write a polynomial function
• Demonstrate the ability to find the zeros of a polynomial function
Conjugate Zeros Theorem Practice Test:

#1:

Instructions: Write a polynomial function of least degree that has the given zeros.

$$a)\hspace{.2em}-1 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, 1 - i$$

$$b)\hspace{.2em}-5, 3i$$

#2:

Instructions: Write a polynomial function of least degree that has the given zeros.

$$a)\hspace{.2em}1, -1, 2 - 2i$$

$$b)\hspace{.2em}4, 2 + i$$

#3:

Instructions: Write a polynomial function of least degree that has the given zeros.

$$a)\hspace{.2em}3 + i \hspace{.25em}\text{multiplicity}\hspace{.25em}2, 3 - i$$

Instructions: Find all zeros.

$$b)\hspace{.2em}f(x)=x^4 - 4x^3 - x^2 - 114x - 102$$ $$\text{zero}:-1 + 4i$$

#4:

Instructions: Find all zeros.

$$a)\hspace{.2em}f(x)=x^3 - 7x^2 + 17x - 15$$ $$\text{zero}:3$$

$$b)\hspace{.2em}f(x)=x^3 - 8x^2 + 6x + 52$$ $$\text{zero}:5 - i$$

#5:

Instructions: Find all zeros.

$$a)\hspace{.2em}f(x)=x^4 - 4x^3 + x^2 + 6x - 40$$ $$\text{zero}:1 + 2i$$

$$b)\hspace{.2em}f(x)=x^4 + 2x^3 - 22x^2 - 30x + 153$$ $$\text{zero}:-4 + i$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}f(x)=x^4 - x^2 + 2x + 2$$

$$b)\hspace{.2em}f(x)=x^3 + 5x^2 + 9x + 45$$

#2:

Solutions:

$$a)\hspace{.2em}f(x)=x^4 - 4x^3 + 7x^2 + 4x - 8$$

$$b)\hspace{.2em}f(x)=x^3 - 8x^2 + 21x - 20$$

#3:

Solutions:

$$a)\hspace{.2em}f(x)=x^4 - 12x^3 + 56x^2 - 120x + 100$$

$$b)\hspace{.2em}-1 \pm 4i, 3 \pm \sqrt{15}$$

#4:

Solutions:

$$a)\hspace{.2em}3, 2 \pm i$$

$$b)\hspace{.2em}-2, 5 \pm i$$

#5:

Solutions:

$$a)\hspace{.2em}-2, 4, 1 \pm 2i$$

$$b)\hspace{.2em}3, -4 \pm i$$