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
#1:
Instructions: Write a polynomial function of least degree that has the given zeros.
$$a)\hspace{.2em}{-}1 \hspace{.25em}\text{multiplicity}\hspace{.25em}2, 1 - i$$
$$b)\hspace{.2em}{-}5, 3i$$
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#2:
Instructions: Write a polynomial function of least degree that has the given zeros.
$$a)\hspace{.2em}1, -1, 2 - 2i$$
$$b)\hspace{.2em}4, 2 + i$$
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#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.
$$b)\hspace{.2em}f(x)=x^4 - 4x^3 - x^2 - 114x - 102$$ $$\text{zero}:-1 + 4i$$
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#4:
Instructions: Find all zeros.
$$a)\hspace{.2em}f(x)=x^3 - 7x^2 + 17x - 15$$ $$\text{zero}:3$$
$$b)\hspace{.2em}f(x)=x^3 - 8x^2 + 6x + 52$$ $$\text{zero}:5 - i$$
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#5:
Instructions: Find all zeros.
$$a)\hspace{.2em}f(x)=x^4 - 4x^3 + x^2 + 6x - 40$$ $$\text{zero}:1 + 2i$$
$$b)\hspace{.2em}f(x)=x^4 + 2x^3 - 22x^2 - 30x + 153$$ $$\text{zero}:-4 + i$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}f(x)=x^4 - x^2 + 2x + 2$$
$$b)\hspace{.2em}f(x)=x^3 + 5x^2 + 9x + 45$$
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#2:
Solutions:
$$a)\hspace{.2em}f(x)=x^4 - 4x^3 + 7x^2 + 4x - 8$$
$$b)\hspace{.2em}f(x)=x^3 - 8x^2 + 21x - 20$$
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#3:
Solutions:
$$a)\hspace{.2em}f(x)=x^4 - 12x^3 + 56x^2 - 120x + 100$$
$$b)\hspace{.2em}{-}1 \pm 4i, 3 \pm \sqrt{15}$$
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#4:
Solutions:
$$a)\hspace{.2em}3, 2 \pm i$$
$$b)\hspace{.2em}{-}2, 5 \pm i$$
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#5:
Solutions:
$$a)\hspace{.2em}{-}2, 4, 1 \pm 2i$$
$$b)\hspace{.2em}3, -4 \pm i$$
